Find the Maximum/Minimum Value x^3+3x^2-x-3

$x^{3} + 3 x^{2} - x - 3$

Find the first derivative of the function.

Differentiate.

By the Sum Rule, the derivative of $x^{3} + 3 x^{2} - x - 3$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 3]$ .

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 3]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 3]$

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$3 x^{2} + 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 3]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$3 x^{2} + 3 (2 x) + \frac{d}{d x} [- x] + \frac{d}{d x} [- 3]$

Multiply $2$ by $3$ .

$3 x^{2} + 6 x + \frac{d}{d x} [- x] + \frac{d}{d x} [- 3]$

Evaluate $\frac{d}{d x} [- x]$ .

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$3 x^{2} + 6 x - \frac{d}{d x} [x] + \frac{d}{d x} [- 3]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$3 x^{2} + 6 x - 1 \cdot 1 + \frac{d}{d x} [- 3]$

Multiply $- 1$ by $1$ .

$3 x^{2} + 6 x - 1 + \frac{d}{d x} [- 3]$

Differentiate using the Constant Rule.

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$3 x^{2} + 6 x - 1 + 0$

Add $3 x^{2} + 6 x - 1$ and $0$ .

$3 x^{2} + 6 x - 1$

Find the second derivative of the function.

By the Sum Rule, the derivative of $3 x^{2} + 6 x - 1$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 1]$ .

$f'' (x) = \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (6 x) + \frac{d}{d x} (- 1)$

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 3 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (6 x) + \frac{d}{d x} (- 1)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f'' (x) = 3 (2 x) + \frac{d}{d x} (6 x) + \frac{d}{d x} (- 1)$

Multiply $2$ by $3$ .

$f'' (x) = 6 x + \frac{d}{d x} (6 x) + \frac{d}{d x} (- 1)$

Evaluate $\frac{d}{d x} [6 x]$ .

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$f'' (x) = 6 x + 6 \frac{d}{d x} (x) + \frac{d}{d x} (- 1)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = 6 x + 6 \cdot 1 + \frac{d}{d x} (- 1)$

Multiply $6$ by $1$ .

$f'' (x) = 6 x + 6 + \frac{d}{d x} (- 1)$

Differentiate using the Constant Rule.

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$f'' (x) = 6 x + 6 + 0$

Add $6 x + 6$ and $0$ .

$f'' (x) = 6 x + 6$

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$3 x^{2} + 6 x - 1 = 0$

Find the first derivative.

Differentiate.

By the Sum Rule, the derivative of $x^{3} + 3 x^{2} - x - 3$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- x] + \frac{d}{d x} [- 3]$ .

$f' (x) = \frac{d}{d x} (x^{3}) + \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- x) + \frac{d}{d x} (- 3)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f' (x) = 3 x^{2} + \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- x) + \frac{d}{d x} (- 3)$

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 3 x^{2} + 3 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- x) + \frac{d}{d x} (- 3)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f' (x) = 3 x^{2} + 3 (2 x) + \frac{d}{d x} (- x) + \frac{d}{d x} (- 3)$

Multiply $2$ by $3$ .

$f' (x) = 3 x^{2} + 6 x + \frac{d}{d x} (- x) + \frac{d}{d x} (- 3)$

Evaluate $\frac{d}{d x} [- x]$ .

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$f' (x) = 3 x^{2} + 6 x - \frac{d}{d x} x + \frac{d}{d x} (- 3)$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = 3 x^{2} + 6 x - 1 \cdot 1 + \frac{d}{d x} (- 3)$

Multiply $- 1$ by $1$ .

$f' (x) = 3 x^{2} + 6 x - 1 + \frac{d}{d x} (- 3)$

Differentiate using the Constant Rule.

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$f' (x) = 3 x^{2} + 6 x - 1 + 0$

Add $3 x^{2} + 6 x - 1$ and $0$ .

$f' (x) = 3 x^{2} + 6 x - 1$

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2} + 6 x - 1$ .

$3 x^{2} + 6 x - 1$

Set the first derivative equal to $0$ then solve the equation $3 x^{2} + 6 x - 1 = 0$ .

Set the first derivative equal to $0$ .

$3 x^{2} + 6 x - 1 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 3$ , $b = 6$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (3 \cdot - 1)}}{2 \cdot 3}$

Simplify.

Simplify the numerator.

Raise $6$ to the power of $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 4 \cdot 3 \cdot - 1}}{2 \cdot 3}$

Multiply $- 4$ by $3$ .

$x = \frac{- 6 \pm \sqrt{36 - 12 \cdot - 1}}{2 \cdot 3}$

Multiply $- 12$ by $- 1$ .

$x = \frac{- 6 \pm \sqrt{36 + 12}}{2 \cdot 3}$

Add $36$ and $12$ .

$x = \frac{- 6 \pm \sqrt{48}}{2 \cdot 3}$

Rewrite $48$ as $4^{2} \cdot 3$ .

Factor $16$ out of $48$ .

$x = \frac{- 6 \pm \sqrt{16 (3)}}{2 \cdot 3}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{- 6 \pm \sqrt{4^{2} \cdot 3}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{- 6 \pm 4 \sqrt{3}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{- 6 \pm 4 \sqrt{3}}{6}$

Simplify $\frac{- 6 \pm 4 \sqrt{3}}{6}$ .

$x = \frac{- 3 \pm 2 \sqrt{3}}{3}$

The final answer is the combination of both solutions.

$x = - \frac{3 - 2 \sqrt{3}}{3}, - \frac{3 + 2 \sqrt{3}}{3}$

Find the values where the derivative is undefined.

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Critical points to evaluate.

$x = - \frac{3 - 2 \sqrt{3}}{3}, - \frac{3 + 2 \sqrt{3}}{3}$

Evaluate the second derivative at $x = - \frac{3 - 2 \sqrt{3}}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$6 (- \frac{3 - 2 \sqrt{3}}{3}) + 6$

Evaluate the second derivative.

Simplify each term.

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{3 - 2 \sqrt{3}}{3}$ into the numerator.

$6 \frac{- (3 - 2 \sqrt{3})}{3} + 6$

Factor $3$ out of $6$ .

$3 (2) \frac{- (3 - 2 \sqrt{3})}{3} + 6$

Cancel the common factor.

$3 \cdot 2 \frac{- (3 - 2 \sqrt{3})}{3} + 6$

Rewrite the expression.

$2 (- (3 - 2 \sqrt{3})) + 6$

Multiply $- 1$ by $2$ .

$- 2 (3 - 2 \sqrt{3}) + 6$

Apply the distributive property.

$- 2 \cdot 3 - 2 (- 2 \sqrt{3}) + 6$

Multiply $- 2$ by $3$ .

$- 6 - 2 (- 2 \sqrt{3}) + 6$

Multiply $- 2$ by $- 2$ .

$- 6 + 4 \sqrt{3} + 6$

Simplify by adding numbers.

Add $- 6$ and $6$ .

$0 + 4 \sqrt{3}$

Add $0$ and $4 \sqrt{3}$ .

$4 \sqrt{3}$

$x = - \frac{3 - 2 \sqrt{3}}{3}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = - \frac{3 - 2 \sqrt{3}}{3}$ is a local minimum

Find the y-value when $x = - \frac{3 - 2 \sqrt{3}}{3}$ .

Replace the variable $x$ with $- \frac{3 - 2 \sqrt{3}}{3}$ in the expression.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = {(- \frac{3 - 2 \sqrt{3}}{3})}^{3} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Simplify the result.

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{3 - 2 \sqrt{3}}{3}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = {(- 1)}^{3} {(\frac{3 - 2 \sqrt{3}}{3})}^{3} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Apply the product rule to $\frac{3 - 2 \sqrt{3}}{3}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = {(- 1)}^{3} (\frac{{(3 - 2 \sqrt{3})}^{3}}{3^{3}}) + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Raise $- 1$ to the power of $3$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{{(3 - 2 \sqrt{3})}^{3}}{3^{3}} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Raise $3$ to the power of $3$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{{(3 - 2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Use the Binomial Theorem.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{3^{3} + 3 \cdot (3^{2} (- 2 \sqrt{3})) + 3 \cdot (3 {(- 2 \sqrt{3})}^{2}) + {(- 2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Simplify each term.

Raise $3$ to the power of $3$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 + 3 \cdot (3^{2} (- 2 \sqrt{3})) + 3 \cdot (3 {(- 2 \sqrt{3})}^{2}) + {(- 2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Multiply $3$ by $3^{2}$ by adding the exponents.

Multiply $3$ by $3^{2}$ .

Raise $3$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 + 3^{1 + 2} (- 2 \sqrt{3}) + 3 \cdot (3 {(- 2 \sqrt{3})}^{2}) + {(- 2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Add $1$ and $2$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 + 3^{3} (- 2 \sqrt{3}) + 3 \cdot (3 {(- 2 \sqrt{3})}^{2}) + {(- 2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Raise $3$ to the power of $3$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 + 27 (- 2 \sqrt{3}) + 3 \cdot (3 {(- 2 \sqrt{3})}^{2}) + {(- 2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Multiply $- 2$ by $27$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 - 54 \sqrt{3} + 3 \cdot (3 {(- 2 \sqrt{3})}^{2}) + {(- 2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Multiply $3$ by $3$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 - 54 \sqrt{3} + 9 {(- 2 \sqrt{3})}^{2} + {(- 2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Apply the product rule to $- 2 \sqrt{3}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 - 54 \sqrt{3} + 9 ({(- 2)}^{2} {\sqrt{3}}^{2}) + {(- 2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Raise $- 2$ to the power of $2$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 - 54 \sqrt{3} + 9 (4 {\sqrt{3}}^{2}) + {(- 2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 - 54 \sqrt{3} + 9 (4 {(3^{\frac{1}{2}})}^{2}) + {(- 2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 - 54 \sqrt{3} + 9 (4 \cdot 3^{\frac{1}{2} \cdot 2}) + {(- 2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 - 54 \sqrt{3} + 9 (4 \cdot 3^{\frac{2}{2}}) + {(- 2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 - 54 \sqrt{3} + 9 (4 \cdot 3^{\frac{2}{2}}) + {(- 2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Rewrite the expression.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 - 54 \sqrt{3} + 9 (4 \cdot 3) + {(- 2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Evaluate the exponent.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 - 54 \sqrt{3} + 9 (4 \cdot 3) + {(- 2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Multiply $4$ by $3$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 - 54 \sqrt{3} + 9 \cdot 12 + {(- 2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Multiply $9$ by $12$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 - 54 \sqrt{3} + 108 + {(- 2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Apply the product rule to $- 2 \sqrt{3}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 - 54 \sqrt{3} + 108 + {(- 2)}^{3} {\sqrt{3}}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Raise $- 2$ to the power of $3$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 - 54 \sqrt{3} + 108 - 8 {\sqrt{3}}^{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Rewrite ${\sqrt{3}}^{3}$ as ${(3^{3})}^{\frac{1}{2}}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 - 54 \sqrt{3} + 108 - 8 \sqrt{3^{3}}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Raise $3$ to the power of $3$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 - 54 \sqrt{3} + 108 - 8 \sqrt{27}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 - 54 \sqrt{3} + 108 - 8 \sqrt{9 (3)}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Rewrite $9$ as $3^{2}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 - 54 \sqrt{3} + 108 - 8 \sqrt{3^{2} \cdot 3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Pull terms out from under the radical.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 - 54 \sqrt{3} + 108 - 8 (3 \sqrt{3})}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Multiply $3$ by $- 8$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{27 - 54 \sqrt{3} + 108 - 24 \sqrt{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Add $27$ and $108$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{135 - 54 \sqrt{3} - 24 \sqrt{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Subtract $24 \sqrt{3}$ from $- 54 \sqrt{3}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{135 - 78 \sqrt{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Cancel the common factor of $135 - 78 \sqrt{3}$ and $27$ .

Factor $3$ out of $135$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{3 (45) - 78 \sqrt{3}}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Factor $3$ out of $- 78 \sqrt{3}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{3 (45) + 3 (- 26 \sqrt{3})}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Factor $3$ out of $3 (45) + 3 (- 26 \sqrt{3})$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{3 (45 - 26 \sqrt{3})}{27} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Cancel the common factors.

Factor $3$ out of $27$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{3 (45 - 26 \sqrt{3})}{3 \cdot 9} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Cancel the common factor.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{3 (45 - 26 \sqrt{3})}{3 \cdot 9} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Rewrite the expression.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + 3 {(- \frac{3 - 2 \sqrt{3}}{3})}^{2} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{3 - 2 \sqrt{3}}{3}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + 3 ({(- 1)}^{2} {(\frac{3 - 2 \sqrt{3}}{3})}^{2}) - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Apply the product rule to $\frac{3 - 2 \sqrt{3}}{3}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + 3 ({(- 1)}^{2} (\frac{{(3 - 2 \sqrt{3})}^{2}}{3^{2}})) - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Raise $- 1$ to the power of $2$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + 3 (1 (\frac{{(3 - 2 \sqrt{3})}^{2}}{3^{2}})) - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Multiply $\frac{{(3 - 2 \sqrt{3})}^{2}}{3^{2}}$ by $1$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + 3 (\frac{{(3 - 2 \sqrt{3})}^{2}}{3^{2}}) - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Raise $3$ to the power of $2$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + 3 (\frac{{(3 - 2 \sqrt{3})}^{2}}{9}) - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + 3 (\frac{{(3 - 2 \sqrt{3})}^{2}}{3 (3)}) - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Cancel the common factor.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + 3 (\frac{{(3 - 2 \sqrt{3})}^{2}}{3 \cdot 3}) - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Rewrite the expression.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{{(3 - 2 \sqrt{3})}^{2}}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Rewrite ${(3 - 2 \sqrt{3})}^{2}$ as $(3 - 2 \sqrt{3}) (3 - 2 \sqrt{3})$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{(3 - 2 \sqrt{3}) (3 - 2 \sqrt{3})}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Expand $(3 - 2 \sqrt{3}) (3 - 2 \sqrt{3})$ using the FOIL Method.

Apply the distributive property.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{3 (3 - 2 \sqrt{3}) - 2 \sqrt{3} (3 - 2 \sqrt{3})}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Apply the distributive property.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{3 \cdot 3 + 3 (- 2 \sqrt{3}) - 2 \sqrt{3} (3 - 2 \sqrt{3})}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Apply the distributive property.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{3 \cdot 3 + 3 (- 2 \sqrt{3}) - 2 \sqrt{3} \cdot 3 - 2 \sqrt{3} (- 2 \sqrt{3})}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{9 + 3 (- 2 \sqrt{3}) - 2 \sqrt{3} \cdot 3 - 2 \sqrt{3} (- 2 \sqrt{3})}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Multiply $- 2$ by $3$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{9 - 6 \sqrt{3} - 2 \sqrt{3} \cdot 3 - 2 \sqrt{3} (- 2 \sqrt{3})}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Multiply $3$ by $- 2$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{9 - 6 \sqrt{3} - 6 \sqrt{3} - 2 \sqrt{3} (- 2 \sqrt{3})}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Multiply $- 2 \sqrt{3} (- 2 \sqrt{3})$ .

Multiply $- 2$ by $- 2$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{9 - 6 \sqrt{3} - 6 \sqrt{3} + 4 \sqrt{3} \sqrt{3}}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Raise $\sqrt{3}$ to the power of $1$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{9 - 6 \sqrt{3} - 6 \sqrt{3} + 4 (\sqrt{3} \sqrt{3})}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Raise $\sqrt{3}$ to the power of $1$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{9 - 6 \sqrt{3} - 6 \sqrt{3} + 4 (\sqrt{3} \sqrt{3})}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{9 - 6 \sqrt{3} - 6 \sqrt{3} + 4 {\sqrt{3}}^{1 + 1}}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Add $1$ and $1$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{9 - 6 \sqrt{3} - 6 \sqrt{3} + 4 {\sqrt{3}}^{2}}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{9 - 6 \sqrt{3} - 6 \sqrt{3} + 4 {(3^{\frac{1}{2}})}^{2}}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{9 - 6 \sqrt{3} - 6 \sqrt{3} + 4 \cdot 3^{\frac{1}{2} \cdot 2}}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{9 - 6 \sqrt{3} - 6 \sqrt{3} + 4 \cdot 3^{\frac{2}{2}}}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{9 - 6 \sqrt{3} - 6 \sqrt{3} + 4 \cdot 3^{\frac{2}{2}}}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Rewrite the expression.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{9 - 6 \sqrt{3} - 6 \sqrt{3} + 4 \cdot 3}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Evaluate the exponent.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{9 - 6 \sqrt{3} - 6 \sqrt{3} + 4 \cdot 3}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Multiply $4$ by $3$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{9 - 6 \sqrt{3} - 6 \sqrt{3} + 12}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Add $9$ and $12$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{21 - 6 \sqrt{3} - 6 \sqrt{3}}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Subtract $6 \sqrt{3}$ from $- 6 \sqrt{3}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{21 - 12 \sqrt{3}}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Cancel the common factor of $21 - 12 \sqrt{3}$ and $3$ .

Factor $3$ out of $21$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{3 \cdot 7 - 12 \sqrt{3}}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Factor $3$ out of $- 12 \sqrt{3}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{3 \cdot 7 + 3 (- 4 \sqrt{3})}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Factor $3$ out of $3 (7) + 3 (- 4 \sqrt{3})$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{3 (7 - 4 \sqrt{3})}{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Cancel the common factors.

Factor $3$ out of $3$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{3 (7 - 4 \sqrt{3})}{3 (1)} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Cancel the common factor.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{3 (7 - 4 \sqrt{3})}{3 \cdot 1} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Rewrite the expression.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{7 - 4 \sqrt{3}}{1} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Divide $7 - 4 \sqrt{3}$ by $1$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + 7 - 4 \sqrt{3} - (- \frac{3 - 2 \sqrt{3}}{3}) - 3$

Multiply $- (- \frac{3 - 2 \sqrt{3}}{3})$ .

Multiply $- 1$ by $- 1$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + 7 - 4 \sqrt{3} + 1 (\frac{3 - 2 \sqrt{3}}{3}) - 3$

Multiply $\frac{3 - 2 \sqrt{3}}{3}$ by $1$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + 7 - 4 \sqrt{3} + \frac{3 - 2 \sqrt{3}}{3} - 3$

To write $7$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + 7 \cdot \frac{9}{9} - 4 \sqrt{3} + \frac{3 - 2 \sqrt{3}}{3} - 3$

Combine fractions.

Combine $7$ and $\frac{9}{9}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{45 - 26 \sqrt{3}}{9} + \frac{7 \cdot 9}{9} - 4 \sqrt{3} + \frac{3 - 2 \sqrt{3}}{3} - 3$

Simplify the expression.

Combine the numerators over the common denominator.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{- (45 - 26 \sqrt{3}) + 7 \cdot 9}{9} - 4 \sqrt{3} + \frac{3 - 2 \sqrt{3}}{3} - 3$

Multiply $7$ by $9$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{- (45 - 26 \sqrt{3}) + 63}{9} - 4 \sqrt{3} + \frac{3 - 2 \sqrt{3}}{3} - 3$

Simplify the numerator.

Apply the distributive property.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{- 1 \cdot 45 - (- 26 \sqrt{3}) + 63}{9} - 4 \sqrt{3} + \frac{3 - 2 \sqrt{3}}{3} - 3$

Multiply $- 1$ by $45$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{- 45 - (- 26 \sqrt{3}) + 63}{9} - 4 \sqrt{3} + \frac{3 - 2 \sqrt{3}}{3} - 3$

Multiply $- 26$ by $- 1$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{- 45 + 26 \sqrt{3} + 63}{9} - 4 \sqrt{3} + \frac{3 - 2 \sqrt{3}}{3} - 3$

Add $- 45$ and $63$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{18 + 26 \sqrt{3}}{9} - 4 \sqrt{3} + \frac{3 - 2 \sqrt{3}}{3} - 3$

Find the common denominator.

Write $- 4 \sqrt{3}$ as a fraction with denominator $1$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{18 + 26 \sqrt{3}}{9} + \frac{- 4 \sqrt{3}}{1} + \frac{3 - 2 \sqrt{3}}{3} - 3$

Multiply $\frac{- 4 \sqrt{3}}{1}$ by $\frac{9}{9}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{18 + 26 \sqrt{3}}{9} + \frac{- 4 \sqrt{3}}{1} \cdot \frac{9}{9} + \frac{3 - 2 \sqrt{3}}{3} - 3$

Multiply $\frac{- 4 \sqrt{3}}{1}$ by $\frac{9}{9}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{18 + 26 \sqrt{3}}{9} + \frac{- 4 \sqrt{3} \cdot 9}{9} + \frac{3 - 2 \sqrt{3}}{3} - 3$

Multiply $\frac{3 - 2 \sqrt{3}}{3}$ by $\frac{3}{3}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{18 + 26 \sqrt{3}}{9} + \frac{- 4 \sqrt{3} \cdot 9}{9} + \frac{3 - 2 \sqrt{3}}{3} \cdot \frac{3}{3} - 3$

Multiply $\frac{3 - 2 \sqrt{3}}{3}$ by $\frac{3}{3}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{18 + 26 \sqrt{3}}{9} + \frac{- 4 \sqrt{3} \cdot 9}{9} + \frac{(3 - 2 \sqrt{3}) \cdot 3}{3 \cdot 3} - 3$

Write $- 3$ as a fraction with denominator $1$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{18 + 26 \sqrt{3}}{9} + \frac{- 4 \sqrt{3} \cdot 9}{9} + \frac{(3 - 2 \sqrt{3}) \cdot 3}{3 \cdot 3} + \frac{- 3}{1}$

Multiply $\frac{- 3}{1}$ by $\frac{9}{9}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{18 + 26 \sqrt{3}}{9} + \frac{- 4 \sqrt{3} \cdot 9}{9} + \frac{(3 - 2 \sqrt{3}) \cdot 3}{3 \cdot 3} + \frac{- 3}{1} \cdot \frac{9}{9}$

Multiply $\frac{- 3}{1}$ by $\frac{9}{9}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{18 + 26 \sqrt{3}}{9} + \frac{- 4 \sqrt{3} \cdot 9}{9} + \frac{(3 - 2 \sqrt{3}) \cdot 3}{3 \cdot 3} + \frac{- 3 \cdot 9}{9}$

Multiply $3$ by $3$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{18 + 26 \sqrt{3}}{9} + \frac{- 4 \sqrt{3} \cdot 9}{9} + \frac{(3 - 2 \sqrt{3}) \cdot 3}{9} + \frac{- 3 \cdot 9}{9}$

Combine the numerators over the common denominator.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{18 + 26 \sqrt{3} - 4 \sqrt{3} \cdot 9 + (3 - 2 \sqrt{3}) \cdot 3 - 3 \cdot 9}{9}$

Simplify each term.

Multiply $9$ by $- 4$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{18 + 26 \sqrt{3} - 36 \sqrt{3} + (3 - 2 \sqrt{3}) \cdot 3 - 3 \cdot 9}{9}$

Apply the distributive property.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{18 + 26 \sqrt{3} - 36 \sqrt{3} + 3 \cdot 3 - 2 \sqrt{3} \cdot 3 - 3 \cdot 9}{9}$

Multiply $3$ by $3$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{18 + 26 \sqrt{3} - 36 \sqrt{3} + 9 - 2 \sqrt{3} \cdot 3 - 3 \cdot 9}{9}$

Multiply $3$ by $- 2$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{18 + 26 \sqrt{3} - 36 \sqrt{3} + 9 - 6 \sqrt{3} - 3 \cdot 9}{9}$

Multiply $- 3$ by $9$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{18 + 26 \sqrt{3} - 36 \sqrt{3} + 9 - 6 \sqrt{3} - 27}{9}$

Simplify by adding terms.

Add $18$ and $9$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{27 + 26 \sqrt{3} - 36 \sqrt{3} - 6 \sqrt{3} - 27}{9}$

Simplify by subtracting numbers.

Subtract $27$ from $27$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{0 + 26 \sqrt{3} - 36 \sqrt{3} - 6 \sqrt{3}}{9}$

Add $0$ and $26 \sqrt{3}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{26 \sqrt{3} - 36 \sqrt{3} - 6 \sqrt{3}}{9}$

Subtract $36 \sqrt{3}$ from $26 \sqrt{3}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{- 10 \sqrt{3} - 6 \sqrt{3}}{9}$

Subtract $6 \sqrt{3}$ from $- 10 \sqrt{3}$ .

$f (- \frac{3 - 2 \sqrt{3}}{3}) = \frac{- 16 \sqrt{3}}{9}$

Move the negative in front of the fraction.

$f (- \frac{3 - 2 \sqrt{3}}{3}) = - \frac{16 \sqrt{3}}{9}$

The final answer is $- \frac{16 \sqrt{3}}{9}$ .

$y = - \frac{16 \sqrt{3}}{9}$

Evaluate the second derivative at $x = - \frac{3 + 2 \sqrt{3}}{3}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$6 (- \frac{3 + 2 \sqrt{3}}{3}) + 6$

Evaluate the second derivative.

Simplify each term.

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{3 + 2 \sqrt{3}}{3}$ into the numerator.

$6 \frac{- (3 + 2 \sqrt{3})}{3} + 6$

Factor $3$ out of $6$ .

$3 (2) \frac{- (3 + 2 \sqrt{3})}{3} + 6$

Cancel the common factor.

$3 \cdot 2 \frac{- (3 + 2 \sqrt{3})}{3} + 6$

Rewrite the expression.

$2 (- (3 + 2 \sqrt{3})) + 6$

Multiply $- 1$ by $2$ .

$- 2 (3 + 2 \sqrt{3}) + 6$

Apply the distributive property.

$- 2 \cdot 3 - 2 (2 \sqrt{3}) + 6$

Multiply $- 2$ by $3$ .

$- 6 - 2 (2 \sqrt{3}) + 6$

Multiply $2$ by $- 2$ .

$- 6 - 4 \sqrt{3} + 6$

Simplify by adding numbers.

Add $- 6$ and $6$ .

$0 - 4 \sqrt{3}$

Subtract $4 \sqrt{3}$ from $0$ .

$- 4 \sqrt{3}$

$x = - \frac{3 + 2 \sqrt{3}}{3}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - \frac{3 + 2 \sqrt{3}}{3}$ is a local maximum

Find the y-value when $x = - \frac{3 + 2 \sqrt{3}}{3}$ .

Replace the variable $x$ with $- \frac{3 + 2 \sqrt{3}}{3}$ in the expression.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = {(- \frac{3 + 2 \sqrt{3}}{3})}^{3} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Simplify the result.

Simplify each term.

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{3 + 2 \sqrt{3}}{3}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = {(- 1)}^{3} {(\frac{3 + 2 \sqrt{3}}{3})}^{3} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Apply the product rule to $\frac{3 + 2 \sqrt{3}}{3}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = {(- 1)}^{3} (\frac{{(3 + 2 \sqrt{3})}^{3}}{3^{3}}) + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Raise $- 1$ to the power of $3$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{{(3 + 2 \sqrt{3})}^{3}}{3^{3}} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Raise $3$ to the power of $3$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{{(3 + 2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Use the Binomial Theorem.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{3^{3} + 3 \cdot (3^{2} (2 \sqrt{3})) + 3 \cdot (3 {(2 \sqrt{3})}^{2}) + {(2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Simplify each term.

Raise $3$ to the power of $3$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 3 \cdot (3^{2} (2 \sqrt{3})) + 3 \cdot (3 {(2 \sqrt{3})}^{2}) + {(2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Multiply $3$ by $3^{2}$ by adding the exponents.

Multiply $3$ by $3^{2}$ .

Raise $3$ to the power of $1$ .

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 3^{1 + 2} (2 \sqrt{3}) + 3 \cdot (3 {(2 \sqrt{3})}^{2}) + {(2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Add $1$ and $2$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 3^{3} (2 \sqrt{3}) + 3 \cdot (3 {(2 \sqrt{3})}^{2}) + {(2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Raise $3$ to the power of $3$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 27 (2 \sqrt{3}) + 3 \cdot (3 {(2 \sqrt{3})}^{2}) + {(2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Multiply $2$ by $27$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 54 \sqrt{3} + 3 \cdot (3 {(2 \sqrt{3})}^{2}) + {(2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Multiply $3$ by $3$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 54 \sqrt{3} + 9 {(2 \sqrt{3})}^{2} + {(2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Apply the product rule to $2 \sqrt{3}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 54 \sqrt{3} + 9 (2^{2} {\sqrt{3}}^{2}) + {(2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Raise $2$ to the power of $2$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 54 \sqrt{3} + 9 (4 {\sqrt{3}}^{2}) + {(2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 54 \sqrt{3} + 9 (4 {(3^{\frac{1}{2}})}^{2}) + {(2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 54 \sqrt{3} + 9 (4 \cdot 3^{\frac{1}{2} \cdot 2}) + {(2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 54 \sqrt{3} + 9 (4 \cdot 3^{\frac{2}{2}}) + {(2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 54 \sqrt{3} + 9 (4 \cdot 3^{\frac{2}{2}}) + {(2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Rewrite the expression.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 54 \sqrt{3} + 9 (4 \cdot 3) + {(2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Evaluate the exponent.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 54 \sqrt{3} + 9 (4 \cdot 3) + {(2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Multiply $4$ by $3$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 54 \sqrt{3} + 9 \cdot 12 + {(2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Multiply $9$ by $12$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 54 \sqrt{3} + 108 + {(2 \sqrt{3})}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Apply the product rule to $2 \sqrt{3}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 54 \sqrt{3} + 108 + 2^{3} {\sqrt{3}}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Raise $2$ to the power of $3$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 54 \sqrt{3} + 108 + 8 {\sqrt{3}}^{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Rewrite ${\sqrt{3}}^{3}$ as ${(3^{3})}^{\frac{1}{2}}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 54 \sqrt{3} + 108 + 8 \sqrt{3^{3}}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Raise $3$ to the power of $3$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 54 \sqrt{3} + 108 + 8 \sqrt{27}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Rewrite $27$ as $3^{2} \cdot 3$ .

Factor $9$ out of $27$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 54 \sqrt{3} + 108 + 8 \sqrt{9 (3)}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Rewrite $9$ as $3^{2}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 54 \sqrt{3} + 108 + 8 \sqrt{3^{2} \cdot 3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Pull terms out from under the radical.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 54 \sqrt{3} + 108 + 8 (3 \sqrt{3})}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Multiply $3$ by $8$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{27 + 54 \sqrt{3} + 108 + 24 \sqrt{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Add $27$ and $108$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{135 + 54 \sqrt{3} + 24 \sqrt{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Add $54 \sqrt{3}$ and $24 \sqrt{3}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{135 + 78 \sqrt{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Cancel the common factor of $135 + 78 \sqrt{3}$ and $27$ .

Factor $3$ out of $135$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{3 (45) + 78 \sqrt{3}}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Factor $3$ out of $78 \sqrt{3}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{3 (45) + 3 (26 \sqrt{3})}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Factor $3$ out of $3 (45) + 3 (26 \sqrt{3})$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{3 (45 + 26 \sqrt{3})}{27} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Cancel the common factors.

Factor $3$ out of $27$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{3 (45 + 26 \sqrt{3})}{3 \cdot 9} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Cancel the common factor.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{3 (45 + 26 \sqrt{3})}{3 \cdot 9} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Rewrite the expression.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + 3 {(- \frac{3 + 2 \sqrt{3}}{3})}^{2} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $- \frac{3 + 2 \sqrt{3}}{3}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + 3 ({(- 1)}^{2} {(\frac{3 + 2 \sqrt{3}}{3})}^{2}) - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Apply the product rule to $\frac{3 + 2 \sqrt{3}}{3}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + 3 ({(- 1)}^{2} (\frac{{(3 + 2 \sqrt{3})}^{2}}{3^{2}})) - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Raise $- 1$ to the power of $2$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + 3 (1 (\frac{{(3 + 2 \sqrt{3})}^{2}}{3^{2}})) - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Multiply $\frac{{(3 + 2 \sqrt{3})}^{2}}{3^{2}}$ by $1$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + 3 (\frac{{(3 + 2 \sqrt{3})}^{2}}{3^{2}}) - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Raise $3$ to the power of $2$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + 3 (\frac{{(3 + 2 \sqrt{3})}^{2}}{9}) - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Cancel the common factor of $3$ .

Factor $3$ out of $9$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + 3 (\frac{{(3 + 2 \sqrt{3})}^{2}}{3 (3)}) - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Cancel the common factor.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + 3 (\frac{{(3 + 2 \sqrt{3})}^{2}}{3 \cdot 3}) - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Rewrite the expression.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{{(3 + 2 \sqrt{3})}^{2}}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Rewrite ${(3 + 2 \sqrt{3})}^{2}$ as $(3 + 2 \sqrt{3}) (3 + 2 \sqrt{3})$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{(3 + 2 \sqrt{3}) (3 + 2 \sqrt{3})}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Expand $(3 + 2 \sqrt{3}) (3 + 2 \sqrt{3})$ using the FOIL Method.

Apply the distributive property.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{3 (3 + 2 \sqrt{3}) + 2 \sqrt{3} (3 + 2 \sqrt{3})}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Apply the distributive property.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{3 \cdot 3 + 3 (2 \sqrt{3}) + 2 \sqrt{3} (3 + 2 \sqrt{3})}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Apply the distributive property.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{3 \cdot 3 + 3 (2 \sqrt{3}) + 2 \sqrt{3} \cdot 3 + 2 \sqrt{3} (2 \sqrt{3})}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{9 + 3 (2 \sqrt{3}) + 2 \sqrt{3} \cdot 3 + 2 \sqrt{3} (2 \sqrt{3})}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Multiply $2$ by $3$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{9 + 6 \sqrt{3} + 2 \sqrt{3} \cdot 3 + 2 \sqrt{3} (2 \sqrt{3})}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Multiply $3$ by $2$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{9 + 6 \sqrt{3} + 6 \sqrt{3} + 2 \sqrt{3} (2 \sqrt{3})}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Multiply $2 \sqrt{3} (2 \sqrt{3})$ .

Multiply $2$ by $2$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{9 + 6 \sqrt{3} + 6 \sqrt{3} + 4 \sqrt{3} \sqrt{3}}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Raise $\sqrt{3}$ to the power of $1$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{9 + 6 \sqrt{3} + 6 \sqrt{3} + 4 (\sqrt{3} \sqrt{3})}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Raise $\sqrt{3}$ to the power of $1$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{9 + 6 \sqrt{3} + 6 \sqrt{3} + 4 (\sqrt{3} \sqrt{3})}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{9 + 6 \sqrt{3} + 6 \sqrt{3} + 4 {\sqrt{3}}^{1 + 1}}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Add $1$ and $1$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{9 + 6 \sqrt{3} + 6 \sqrt{3} + 4 {\sqrt{3}}^{2}}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{9 + 6 \sqrt{3} + 6 \sqrt{3} + 4 {(3^{\frac{1}{2}})}^{2}}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{9 + 6 \sqrt{3} + 6 \sqrt{3} + 4 \cdot 3^{\frac{1}{2} \cdot 2}}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Combine $\frac{1}{2}$ and $2$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{9 + 6 \sqrt{3} + 6 \sqrt{3} + 4 \cdot 3^{\frac{2}{2}}}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Cancel the common factor of $2$ .

Cancel the common factor.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{9 + 6 \sqrt{3} + 6 \sqrt{3} + 4 \cdot 3^{\frac{2}{2}}}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Rewrite the expression.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{9 + 6 \sqrt{3} + 6 \sqrt{3} + 4 \cdot 3}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Evaluate the exponent.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{9 + 6 \sqrt{3} + 6 \sqrt{3} + 4 \cdot 3}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Multiply $4$ by $3$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{9 + 6 \sqrt{3} + 6 \sqrt{3} + 12}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Add $9$ and $12$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{21 + 6 \sqrt{3} + 6 \sqrt{3}}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Add $6 \sqrt{3}$ and $6 \sqrt{3}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{21 + 12 \sqrt{3}}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Cancel the common factor of $21 + 12 \sqrt{3}$ and $3$ .

Factor $3$ out of $21$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{3 \cdot 7 + 12 \sqrt{3}}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Factor $3$ out of $12 \sqrt{3}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{3 \cdot 7 + 3 (4 \sqrt{3})}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Factor $3$ out of $3 (7) + 3 (4 \sqrt{3})$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{3 (7 + 4 \sqrt{3})}{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Cancel the common factors.

Factor $3$ out of $3$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{3 (7 + 4 \sqrt{3})}{3 (1)} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Cancel the common factor.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{3 (7 + 4 \sqrt{3})}{3 \cdot 1} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Rewrite the expression.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{7 + 4 \sqrt{3}}{1} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Divide $7 + 4 \sqrt{3}$ by $1$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + 7 + 4 \sqrt{3} - (- \frac{3 + 2 \sqrt{3}}{3}) - 3$

Multiply $- (- \frac{3 + 2 \sqrt{3}}{3})$ .

Multiply $- 1$ by $- 1$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + 7 + 4 \sqrt{3} + 1 (\frac{3 + 2 \sqrt{3}}{3}) - 3$

Multiply $\frac{3 + 2 \sqrt{3}}{3}$ by $1$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + 7 + 4 \sqrt{3} + \frac{3 + 2 \sqrt{3}}{3} - 3$

To write $7$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + 7 \cdot \frac{9}{9} + 4 \sqrt{3} + \frac{3 + 2 \sqrt{3}}{3} - 3$

Combine fractions.

Combine $7$ and $\frac{9}{9}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = - \frac{45 + 26 \sqrt{3}}{9} + \frac{7 \cdot 9}{9} + 4 \sqrt{3} + \frac{3 + 2 \sqrt{3}}{3} - 3$

Simplify the expression.

Combine the numerators over the common denominator.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{- (45 + 26 \sqrt{3}) + 7 \cdot 9}{9} + 4 \sqrt{3} + \frac{3 + 2 \sqrt{3}}{3} - 3$

Multiply $7$ by $9$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{- (45 + 26 \sqrt{3}) + 63}{9} + 4 \sqrt{3} + \frac{3 + 2 \sqrt{3}}{3} - 3$

Simplify the numerator.

Apply the distributive property.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{- 1 \cdot 45 - (26 \sqrt{3}) + 63}{9} + 4 \sqrt{3} + \frac{3 + 2 \sqrt{3}}{3} - 3$

Multiply $- 1$ by $45$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{- 45 - (26 \sqrt{3}) + 63}{9} + 4 \sqrt{3} + \frac{3 + 2 \sqrt{3}}{3} - 3$

Multiply $26$ by $- 1$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{- 45 - 26 \sqrt{3} + 63}{9} + 4 \sqrt{3} + \frac{3 + 2 \sqrt{3}}{3} - 3$

Add $- 45$ and $63$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{18 - 26 \sqrt{3}}{9} + 4 \sqrt{3} + \frac{3 + 2 \sqrt{3}}{3} - 3$

Find the common denominator.

Write $4 \sqrt{3}$ as a fraction with denominator $1$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{18 - 26 \sqrt{3}}{9} + \frac{4 \sqrt{3}}{1} + \frac{3 + 2 \sqrt{3}}{3} - 3$

Multiply $\frac{4 \sqrt{3}}{1}$ by $\frac{9}{9}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{18 - 26 \sqrt{3}}{9} + \frac{4 \sqrt{3}}{1} \cdot \frac{9}{9} + \frac{3 + 2 \sqrt{3}}{3} - 3$

Multiply $\frac{4 \sqrt{3}}{1}$ by $\frac{9}{9}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{18 - 26 \sqrt{3}}{9} + \frac{4 \sqrt{3} \cdot 9}{9} + \frac{3 + 2 \sqrt{3}}{3} - 3$

Multiply $\frac{3 + 2 \sqrt{3}}{3}$ by $\frac{3}{3}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{18 - 26 \sqrt{3}}{9} + \frac{4 \sqrt{3} \cdot 9}{9} + \frac{3 + 2 \sqrt{3}}{3} \cdot \frac{3}{3} - 3$

Multiply $\frac{3 + 2 \sqrt{3}}{3}$ by $\frac{3}{3}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{18 - 26 \sqrt{3}}{9} + \frac{4 \sqrt{3} \cdot 9}{9} + \frac{(3 + 2 \sqrt{3}) \cdot 3}{3 \cdot 3} - 3$

Write $- 3$ as a fraction with denominator $1$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{18 - 26 \sqrt{3}}{9} + \frac{4 \sqrt{3} \cdot 9}{9} + \frac{(3 + 2 \sqrt{3}) \cdot 3}{3 \cdot 3} + \frac{- 3}{1}$

Multiply $\frac{- 3}{1}$ by $\frac{9}{9}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{18 - 26 \sqrt{3}}{9} + \frac{4 \sqrt{3} \cdot 9}{9} + \frac{(3 + 2 \sqrt{3}) \cdot 3}{3 \cdot 3} + \frac{- 3}{1} \cdot \frac{9}{9}$

Multiply $\frac{- 3}{1}$ by $\frac{9}{9}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{18 - 26 \sqrt{3}}{9} + \frac{4 \sqrt{3} \cdot 9}{9} + \frac{(3 + 2 \sqrt{3}) \cdot 3}{3 \cdot 3} + \frac{- 3 \cdot 9}{9}$

Multiply $3$ by $3$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{18 - 26 \sqrt{3}}{9} + \frac{4 \sqrt{3} \cdot 9}{9} + \frac{(3 + 2 \sqrt{3}) \cdot 3}{9} + \frac{- 3 \cdot 9}{9}$

Combine the numerators over the common denominator.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{18 - 26 \sqrt{3} + 4 \sqrt{3} \cdot 9 + (3 + 2 \sqrt{3}) \cdot 3 - 3 \cdot 9}{9}$

Simplify each term.

Multiply $9$ by $4$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{18 - 26 \sqrt{3} + 36 \sqrt{3} + (3 + 2 \sqrt{3}) \cdot 3 - 3 \cdot 9}{9}$

Apply the distributive property.

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{18 - 26 \sqrt{3} + 36 \sqrt{3} + 3 \cdot 3 + 2 \sqrt{3} \cdot 3 - 3 \cdot 9}{9}$

Multiply $3$ by $3$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{18 - 26 \sqrt{3} + 36 \sqrt{3} + 9 + 2 \sqrt{3} \cdot 3 - 3 \cdot 9}{9}$

Multiply $3$ by $2$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{18 - 26 \sqrt{3} + 36 \sqrt{3} + 9 + 6 \sqrt{3} - 3 \cdot 9}{9}$

Multiply $- 3$ by $9$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{18 - 26 \sqrt{3} + 36 \sqrt{3} + 9 + 6 \sqrt{3} - 27}{9}$

Simplify by adding terms.

Add $18$ and $9$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{27 - 26 \sqrt{3} + 36 \sqrt{3} + 6 \sqrt{3} - 27}{9}$

Simplify by subtracting numbers.

Subtract $27$ from $27$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{0 - 26 \sqrt{3} + 36 \sqrt{3} + 6 \sqrt{3}}{9}$

Subtract $26 \sqrt{3}$ from $0$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{- 26 \sqrt{3} + 36 \sqrt{3} + 6 \sqrt{3}}{9}$

Add $- 26 \sqrt{3}$ and $36 \sqrt{3}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{10 \sqrt{3} + 6 \sqrt{3}}{9}$

Add $10 \sqrt{3}$ and $6 \sqrt{3}$ .

$f (- \frac{3 + 2 \sqrt{3}}{3}) = \frac{16 \sqrt{3}}{9}$

The final answer is $\frac{16 \sqrt{3}}{9}$ .

$y = \frac{16 \sqrt{3}}{9}$

These are the local extrema for $f (x) = x^{3} + 3 x^{2} - x - 3$ .

$(- \frac{3 - 2 \sqrt{3}}{3}, - \frac{16 \sqrt{3}}{9})$ is a local minima

$(- \frac{3 + 2 \sqrt{3}}{3}, \frac{16 \sqrt{3}}{9})$ is a local maxima

Find the Maximum/Minimum Value x^3+3x^2-x-3

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