Find the Local Maxima and Minima f(x)=6x^3-5x+12

$f (x) = 6 x^{3} - 5 x + 12$

Find the first derivative of the function.

By the Sum Rule, the derivative of $6 x^{3} - 5 x + 12$ with respect to $x$ is $\frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [12]$ .

$\frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [12]$

Evaluate $\frac{d}{d x} [6 x^{3}]$ .

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{3}$ with respect to $x$ is $6 \frac{d}{d x} [x^{3}]$ .

$6 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [12]$

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

$6 (3 x^{2}) + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [12]$

Multiply $3$ by $6$ .

$18 x^{2} + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [12]$

Evaluate $\frac{d}{d x} [- 5 x]$ .

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$18 x^{2} - 5 \frac{d}{d x} [x] + \frac{d}{d x} [12]$

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

$18 x^{2} - 5 \cdot 1 + \frac{d}{d x} [12]$

Multiply $- 5$ by $1$ .

$18 x^{2} - 5 + \frac{d}{d x} [12]$

Differentiate using the Constant Rule.

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$18 x^{2} - 5 + 0$

Add $18 x^{2} - 5$ and $0$ .

$18 x^{2} - 5$

Find the second derivative of the function.

By the Sum Rule, the derivative of $18 x^{2} - 5$ with respect to $x$ is $\frac{d}{d x} [18 x^{2}] + \frac{d}{d x} [- 5]$ .

$f'' (x) = \frac{d}{d x} (18 x^{2}) + \frac{d}{d x} (- 5)$

Evaluate $\frac{d}{d x} [18 x^{2}]$ .

Since $18$ is constant with respect to $x$ , the derivative of $18 x^{2}$ with respect to $x$ is $18 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 18 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 5)$

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

$f'' (x) = 18 (2 x) + \frac{d}{d x} (- 5)$

Multiply $2$ by $18$ .

$f'' (x) = 36 x + \frac{d}{d x} (- 5)$

Differentiate using the Constant Rule.

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$f'' (x) = 36 x + 0$

Add $36 x$ and $0$ .

$f'' (x) = 36 x$

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

$18 x^{2} - 5 = 0$

Find the first derivative.

By the Sum Rule, the derivative of $6 x^{3} - 5 x + 12$ with respect to $x$ is $\frac{d}{d x} [6 x^{3}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [12]$ .

$f' (x) = \frac{d}{d x} (6 x^{3}) + \frac{d}{d x} (- 5 x) + \frac{d}{d x} (12)$

Evaluate $\frac{d}{d x} [6 x^{3}]$ .

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{3}$ with respect to $x$ is $6 \frac{d}{d x} [x^{3}]$ .

$f' (x) = 6 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 5 x) + \frac{d}{d x} (12)$

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

$f' (x) = 6 (3 x^{2}) + \frac{d}{d x} (- 5 x) + \frac{d}{d x} (12)$

Multiply $3$ by $6$ .

$f' (x) = 18 x^{2} + \frac{d}{d x} (- 5 x) + \frac{d}{d x} (12)$

Evaluate $\frac{d}{d x} [- 5 x]$ .

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$f' (x) = 18 x^{2} - 5 \frac{d}{d x} x + \frac{d}{d x} (12)$

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

$f' (x) = 18 x^{2} - 5 \cdot 1 + \frac{d}{d x} (12)$

Multiply $- 5$ by $1$ .

$f' (x) = 18 x^{2} - 5 + \frac{d}{d x} (12)$

Differentiate using the Constant Rule.

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$f' (x) = 18 x^{2} - 5 + 0$

Add $18 x^{2} - 5$ and $0$ .

$f' (x) = 18 x^{2} - 5$

The first derivative of $f (x)$ with respect to $x$ is $18 x^{2} - 5$ .

$18 x^{2} - 5$

Set the first derivative equal to $0$ then solve the equation $18 x^{2} - 5 = 0$ .

Set the first derivative equal to $0$ .

$18 x^{2} - 5 = 0$

Add $5$ to both sides of the equation.

$18 x^{2} = 5$

Divide each term in $18 x^{2} = 5$ by $18$ and simplify.

Divide each term in $18 x^{2} = 5$ by $18$ .

$\frac{18 x^{2}}{18} = \frac{5}{18}$

Simplify the left side.

Cancel the common factor of $18$ .

Cancel the common factor.

$\frac{18 x^{2}}{18} = \frac{5}{18}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{5}{18}$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{5}{18}}$

Simplify $\pm \sqrt{\frac{5}{18}}$ .

Rewrite $\sqrt{\frac{5}{18}}$ as $\frac{\sqrt{5}}{\sqrt{18}}$ .

$x = \pm \frac{\sqrt{5}}{\sqrt{18}}$

Simplify the denominator.

Rewrite $18$ as $3^{2} \cdot 2$ .

Factor $9$ out of $18$ .

$x = \pm \frac{\sqrt{5}}{\sqrt{9 (2)}}$

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

$x = \pm \frac{\sqrt{5}}{\sqrt{3^{2} \cdot 2}}$

Pull terms out from under the radical.

$x = \pm \frac{\sqrt{5}}{3 \sqrt{2}}$

Multiply $\frac{\sqrt{5}}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{5}}{3 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{5}}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{5} \sqrt{2}}{3 \sqrt{2} \sqrt{2}}$

Move $\sqrt{2}$ .

$x = \pm \frac{\sqrt{5} \sqrt{2}}{3 (\sqrt{2} \sqrt{2})}$

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{5} \sqrt{2}}{3 ({\sqrt{2}}^{1} \sqrt{2})}$

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{5} \sqrt{2}}{3 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

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

$x = \pm \frac{\sqrt{5} \sqrt{2}}{3 {\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{5} \sqrt{2}}{3 {\sqrt{2}}^{2}}$

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

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

$x = \pm \frac{\sqrt{5} \sqrt{2}}{3 {(2^{\frac{1}{2}})}^{2}}$

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

$x = \pm \frac{\sqrt{5} \sqrt{2}}{3 \cdot 2^{\frac{1}{2} \cdot 2}}$

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

$x = \pm \frac{\sqrt{5} \sqrt{2}}{3 \cdot 2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm \frac{\sqrt{5} \sqrt{2}}{3 \cdot 2^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm \frac{\sqrt{5} \sqrt{2}}{3 \cdot 2^{1}}$

Evaluate the exponent.

$x = \pm \frac{\sqrt{5} \sqrt{2}}{3 \cdot 2}$

Simplify the numerator.

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{5 \cdot 2}}{3 \cdot 2}$

Multiply $5$ by $2$ .

$x = \pm \frac{\sqrt{10}}{3 \cdot 2}$

Multiply $3$ by $2$ .

$x = \pm \frac{\sqrt{10}}{6}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{10}}{6}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{10}}{6}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{10}}{6}, - \frac{\sqrt{10}}{6}$

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.

$x = \frac{\sqrt{10}}{6}, - \frac{\sqrt{10}}{6}$

Evaluate the second derivative at $x = \frac{\sqrt{10}}{6}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$36 (\frac{\sqrt{10}}{6})$

Cancel the common factor of $6$ .

Factor $6$ out of $36$ .

$6 (6) \frac{\sqrt{10}}{6}$

Cancel the common factor.

$6 \cdot 6 \frac{\sqrt{10}}{6}$

Rewrite the expression.

$6 \sqrt{10}$

$x = \frac{\sqrt{10}}{6}$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = \frac{\sqrt{10}}{6}$ is a local minimum

Find the y-value when $x = \frac{\sqrt{10}}{6}$ .

Replace the variable $x$ with $\frac{\sqrt{10}}{6}$ in the expression.

$f (\frac{\sqrt{10}}{6}) = 6 {(\frac{\sqrt{10}}{6})}^{3} - 5 \frac{\sqrt{10}}{6} + 12$

Simplify the result.

Simplify each term.

Apply the product rule to $\frac{\sqrt{10}}{6}$ .

$f (\frac{\sqrt{10}}{6}) = 6 (\frac{{\sqrt{10}}^{3}}{6^{3}}) - 5 \frac{\sqrt{10}}{6} + 12$

Simplify the numerator.

Rewrite ${\sqrt{10}}^{3}$ as ${(10^{3})}^{\frac{1}{2}}$ .

$f (\frac{\sqrt{10}}{6}) = 6 (\frac{\sqrt{10^{3}}}{6^{3}}) - 5 \frac{\sqrt{10}}{6} + 12$

Raise $10$ to the power of $3$ .

$f (\frac{\sqrt{10}}{6}) = 6 (\frac{\sqrt{1000}}{6^{3}}) - 5 \frac{\sqrt{10}}{6} + 12$

Rewrite $1000$ as $10^{2} \cdot 10$ .

Factor $100$ out of $1000$ .

$f (\frac{\sqrt{10}}{6}) = 6 (\frac{\sqrt{100 (10)}}{6^{3}}) - 5 \frac{\sqrt{10}}{6} + 12$

Rewrite $100$ as $10^{2}$ .

$f (\frac{\sqrt{10}}{6}) = 6 (\frac{\sqrt{10^{2} \cdot 10}}{6^{3}}) - 5 \frac{\sqrt{10}}{6} + 12$

Pull terms out from under the radical.

$f (\frac{\sqrt{10}}{6}) = 6 (\frac{10 \sqrt{10}}{6^{3}}) - 5 \frac{\sqrt{10}}{6} + 12$

Raise $6$ to the power of $3$ .

$f (\frac{\sqrt{10}}{6}) = 6 (\frac{10 \sqrt{10}}{216}) - 5 \frac{\sqrt{10}}{6} + 12$

Cancel the common factor of $6$ .

Factor $6$ out of $216$ .

$f (\frac{\sqrt{10}}{6}) = 6 (\frac{10 \sqrt{10}}{6 (36)}) - 5 \frac{\sqrt{10}}{6} + 12$

Cancel the common factor.

$f (\frac{\sqrt{10}}{6}) = 6 (\frac{10 \sqrt{10}}{6 \cdot 36}) - 5 \frac{\sqrt{10}}{6} + 12$

Rewrite the expression.

$f (\frac{\sqrt{10}}{6}) = \frac{10 \sqrt{10}}{36} - 5 \frac{\sqrt{10}}{6} + 12$

Cancel the common factor of $10$ and $36$ .

Factor $2$ out of $10 \sqrt{10}$ .

$f (\frac{\sqrt{10}}{6}) = \frac{2 (5 \sqrt{10})}{36} - 5 \frac{\sqrt{10}}{6} + 12$

Cancel the common factors.

Factor $2$ out of $36$ .

$f (\frac{\sqrt{10}}{6}) = \frac{2 (5 \sqrt{10})}{2 (18)} - 5 \frac{\sqrt{10}}{6} + 12$

Cancel the common factor.

$f (\frac{\sqrt{10}}{6}) = \frac{2 (5 \sqrt{10})}{2 \cdot 18} - 5 \frac{\sqrt{10}}{6} + 12$

Rewrite the expression.

$f (\frac{\sqrt{10}}{6}) = \frac{5 \sqrt{10}}{18} - 5 \frac{\sqrt{10}}{6} + 12$

Combine $- 5$ and $\frac{\sqrt{10}}{6}$ .

$f (\frac{\sqrt{10}}{6}) = \frac{5 \sqrt{10}}{18} + \frac{- 5 \sqrt{10}}{6} + 12$

Move the negative in front of the fraction.

$f (\frac{\sqrt{10}}{6}) = \frac{5 \sqrt{10}}{18} - \frac{5 \sqrt{10}}{6} + 12$

To write $- \frac{5 \sqrt{10}}{6}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (\frac{\sqrt{10}}{6}) = \frac{5 \sqrt{10}}{18} - \frac{5 \sqrt{10}}{6} \cdot \frac{3}{3} + 12$

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5 \sqrt{10}}{6}$ by $\frac{3}{3}$ .

$f (\frac{\sqrt{10}}{6}) = \frac{5 \sqrt{10}}{18} - \frac{5 \sqrt{10} \cdot 3}{6 \cdot 3} + 12$

Multiply $6$ by $3$ .

$f (\frac{\sqrt{10}}{6}) = \frac{5 \sqrt{10}}{18} - \frac{5 \sqrt{10} \cdot 3}{18} + 12$

Combine the numerators over the common denominator.

$f (\frac{\sqrt{10}}{6}) = \frac{5 \sqrt{10} - 5 \sqrt{10} \cdot 3}{18} + 12$

Simplify each term.

Simplify the numerator.

Multiply $3$ by $- 5$ .

$f (\frac{\sqrt{10}}{6}) = \frac{5 \sqrt{10} - 15 \sqrt{10}}{18} + 12$

Subtract $15 \sqrt{10}$ from $5 \sqrt{10}$ .

$f (\frac{\sqrt{10}}{6}) = \frac{- 10 \sqrt{10}}{18} + 12$

Cancel the common factor of $- 10$ and $18$ .

Factor $2$ out of $- 10 \sqrt{10}$ .

$f (\frac{\sqrt{10}}{6}) = \frac{2 (- 5 \sqrt{10})}{18} + 12$

Cancel the common factors.

Factor $2$ out of $18$ .

$f (\frac{\sqrt{10}}{6}) = \frac{2 (- 5 \sqrt{10})}{2 (9)} + 12$

Cancel the common factor.

$f (\frac{\sqrt{10}}{6}) = \frac{2 (- 5 \sqrt{10})}{2 \cdot 9} + 12$

Rewrite the expression.

$f (\frac{\sqrt{10}}{6}) = \frac{- 5 \sqrt{10}}{9} + 12$

Move the negative in front of the fraction.

$f (\frac{\sqrt{10}}{6}) = - \frac{5 \sqrt{10}}{9} + 12$

The final answer is $- \frac{5 \sqrt{10}}{9} + 12$ .

$y = - \frac{5 \sqrt{10}}{9} + 12$

Evaluate the second derivative at $x = - \frac{\sqrt{10}}{6}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$36 (- \frac{\sqrt{10}}{6})$

Evaluate the second derivative.

Cancel the common factor of $6$ .

Move the leading negative in $- \frac{\sqrt{10}}{6}$ into the numerator.

$36 \frac{- \sqrt{10}}{6}$

Factor $6$ out of $36$ .

$6 (6) \frac{- \sqrt{10}}{6}$

Cancel the common factor.

$6 \cdot 6 \frac{- \sqrt{10}}{6}$

Rewrite the expression.

$6 (- \sqrt{10})$

Multiply $- 1$ by $6$ .

$- 6 \sqrt{10}$

$x = - \frac{\sqrt{10}}{6}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - \frac{\sqrt{10}}{6}$ is a local maximum

Find the y-value when $x = - \frac{\sqrt{10}}{6}$ .

Replace the variable $x$ with $- \frac{\sqrt{10}}{6}$ in the expression.

$f (- \frac{\sqrt{10}}{6}) = 6 {(- \frac{\sqrt{10}}{6})}^{3} - 5 (- \frac{\sqrt{10}}{6}) + 12$

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 $- \frac{\sqrt{10}}{6}$ .

$f (- \frac{\sqrt{10}}{6}) = 6 ({(- 1)}^{3} {(\frac{\sqrt{10}}{6})}^{3}) - 5 (- \frac{\sqrt{10}}{6}) + 12$

Apply the product rule to $\frac{\sqrt{10}}{6}$ .

$f (- \frac{\sqrt{10}}{6}) = 6 ({(- 1)}^{3} (\frac{{\sqrt{10}}^{3}}{6^{3}})) - 5 (- \frac{\sqrt{10}}{6}) + 12$

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

$f (- \frac{\sqrt{10}}{6}) = 6 (- \frac{{\sqrt{10}}^{3}}{6^{3}}) - 5 (- \frac{\sqrt{10}}{6}) + 12$

Simplify the numerator.

Rewrite ${\sqrt{10}}^{3}$ as ${(10^{3})}^{\frac{1}{2}}$ .

$f (- \frac{\sqrt{10}}{6}) = 6 (- \frac{\sqrt{10^{3}}}{6^{3}}) - 5 (- \frac{\sqrt{10}}{6}) + 12$

Raise $10$ to the power of $3$ .

$f (- \frac{\sqrt{10}}{6}) = 6 (- \frac{\sqrt{1000}}{6^{3}}) - 5 (- \frac{\sqrt{10}}{6}) + 12$

Rewrite $1000$ as $10^{2} \cdot 10$ .

Factor $100$ out of $1000$ .

$f (- \frac{\sqrt{10}}{6}) = 6 (- \frac{\sqrt{100 (10)}}{6^{3}}) - 5 (- \frac{\sqrt{10}}{6}) + 12$

Rewrite $100$ as $10^{2}$ .

$f (- \frac{\sqrt{10}}{6}) = 6 (- \frac{\sqrt{10^{2} \cdot 10}}{6^{3}}) - 5 (- \frac{\sqrt{10}}{6}) + 12$

Pull terms out from under the radical.

$f (- \frac{\sqrt{10}}{6}) = 6 (- \frac{10 \sqrt{10}}{6^{3}}) - 5 (- \frac{\sqrt{10}}{6}) + 12$

Raise $6$ to the power of $3$ .

$f (- \frac{\sqrt{10}}{6}) = 6 (- \frac{10 \sqrt{10}}{216}) - 5 (- \frac{\sqrt{10}}{6}) + 12$

Cancel the common factor of $6$ .

Move the leading negative in $- \frac{10 \sqrt{10}}{216}$ into the numerator.

$f (- \frac{\sqrt{10}}{6}) = 6 (\frac{- 10 \sqrt{10}}{216}) - 5 (- \frac{\sqrt{10}}{6}) + 12$

Factor $6$ out of $216$ .

$f (- \frac{\sqrt{10}}{6}) = 6 (\frac{- 10 \sqrt{10}}{6 (36)}) - 5 (- \frac{\sqrt{10}}{6}) + 12$

Cancel the common factor.

$f (- \frac{\sqrt{10}}{6}) = 6 (\frac{- 10 \sqrt{10}}{6 \cdot 36}) - 5 (- \frac{\sqrt{10}}{6}) + 12$

Rewrite the expression.

$f (- \frac{\sqrt{10}}{6}) = \frac{- 10 \sqrt{10}}{36} - 5 (- \frac{\sqrt{10}}{6}) + 12$

Cancel the common factor of $- 10$ and $36$ .

Factor $2$ out of $- 10 \sqrt{10}$ .

$f (- \frac{\sqrt{10}}{6}) = \frac{2 (- 5 \sqrt{10})}{36} - 5 (- \frac{\sqrt{10}}{6}) + 12$

Cancel the common factors.

Factor $2$ out of $36$ .

$f (- \frac{\sqrt{10}}{6}) = \frac{2 (- 5 \sqrt{10})}{2 (18)} - 5 (- \frac{\sqrt{10}}{6}) + 12$

Cancel the common factor.

$f (- \frac{\sqrt{10}}{6}) = \frac{2 (- 5 \sqrt{10})}{2 \cdot 18} - 5 (- \frac{\sqrt{10}}{6}) + 12$

Rewrite the expression.

$f (- \frac{\sqrt{10}}{6}) = \frac{- 5 \sqrt{10}}{18} - 5 (- \frac{\sqrt{10}}{6}) + 12$

Move the negative in front of the fraction.

$f (- \frac{\sqrt{10}}{6}) = - \frac{(5) \sqrt{10}}{18} - 5 (- \frac{\sqrt{10}}{6}) + 12$

Multiply $- 5 (- \frac{\sqrt{10}}{6})$ .

Multiply $- 1$ by $- 5$ .

$f (- \frac{\sqrt{10}}{6}) = - \frac{5 \sqrt{10}}{18} + 5 (\frac{\sqrt{10}}{6}) + 12$

Combine $5$ and $\frac{\sqrt{10}}{6}$ .

$f (- \frac{\sqrt{10}}{6}) = - \frac{5 \sqrt{10}}{18} + \frac{5 \sqrt{10}}{6} + 12$

To write $\frac{5 \sqrt{10}}{6}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- \frac{\sqrt{10}}{6}) = - \frac{5 \sqrt{10}}{18} + \frac{5 \sqrt{10}}{6} \cdot \frac{3}{3} + 12$

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5 \sqrt{10}}{6}$ by $\frac{3}{3}$ .

$f (- \frac{\sqrt{10}}{6}) = - \frac{5 \sqrt{10}}{18} + \frac{5 \sqrt{10} \cdot 3}{6 \cdot 3} + 12$

Multiply $6$ by $3$ .

$f (- \frac{\sqrt{10}}{6}) = - \frac{5 \sqrt{10}}{18} + \frac{5 \sqrt{10} \cdot 3}{18} + 12$

Combine the numerators over the common denominator.

$f (- \frac{\sqrt{10}}{6}) = \frac{- 5 \sqrt{10} + 5 \sqrt{10} \cdot 3}{18} + 12$

Simplify each term.

Simplify the numerator.

Multiply $3$ by $5$ .

$f (- \frac{\sqrt{10}}{6}) = \frac{- 5 \sqrt{10} + 15 \sqrt{10}}{18} + 12$

Add $- 5 \sqrt{10}$ and $15 \sqrt{10}$ .

$f (- \frac{\sqrt{10}}{6}) = \frac{10 \sqrt{10}}{18} + 12$

Cancel the common factor of $10$ and $18$ .

Factor $2$ out of $10 \sqrt{10}$ .

$f (- \frac{\sqrt{10}}{6}) = \frac{2 (5 \sqrt{10})}{18} + 12$

Cancel the common factors.

Factor $2$ out of $18$ .

$f (- \frac{\sqrt{10}}{6}) = \frac{2 (5 \sqrt{10})}{2 (9)} + 12$

Cancel the common factor.

$f (- \frac{\sqrt{10}}{6}) = \frac{2 (5 \sqrt{10})}{2 \cdot 9} + 12$

Rewrite the expression.

$f (- \frac{\sqrt{10}}{6}) = \frac{5 \sqrt{10}}{9} + 12$

The final answer is $\frac{5 \sqrt{10}}{9} + 12$ .

$y = \frac{5 \sqrt{10}}{9} + 12$

These are the local extrema for $f (x) = 6 x^{3} - 5 x + 12$ .

$(\frac{\sqrt{10}}{6}, - \frac{5 \sqrt{10}}{9} + 12)$ is a local minima

$(- \frac{\sqrt{10}}{6}, \frac{5 \sqrt{10}}{9} + 12)$ is a local maxima

Find the Local Maxima and Minima f(x)=6x^3-5x+12

Related Questions

Support LCMGCF.com by disabling your adblocker.