Find the Local Maxima and Minima y=1/3x^3-5/2x^2+6x-3

$y = \frac{1}{3} x^{3} - \frac{5}{2} x^{2} + 6 x - 3$

Step 1

Write $y = \frac{1}{3} x^{3} - \frac{5}{2} x^{2} + 6 x - 3$ as a function.

$f (x) = \frac{1}{3} x^{3} - \frac{5}{2} x^{2} + 6 x - 3$

Step 2

Find the first derivative of the function.

By the Sum Rule, the derivative of $\frac{1}{3} x^{3} - \frac{5}{2} x^{2} + 6 x - 3$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [- \frac{5}{2} x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 3]$ .

$\frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [- \frac{5}{2} x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 3]$

Evaluate $\frac{d}{d x} [\frac{1}{3} x^{3}]$ .

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{1}{3} x^{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x^{3}]$ .

$\frac{1}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- \frac{5}{2} x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 3]$

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

$\frac{1}{3} (3 x^{2}) + \frac{d}{d x} [- \frac{5}{2} x^{2}] + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 3]$

Combine $3$ and $\frac{1}{3}$ .

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

Combine $\frac{3}{3}$ and $x^{2}$ .

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

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

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

Evaluate $\frac{d}{d x} [- \frac{5}{2} x^{2}]$ .

Since $- \frac{5}{2}$ is constant with respect to $x$ , the derivative of $- \frac{5}{2} x^{2}$ with respect to $x$ is $- \frac{5}{2} \frac{d}{d x} [x^{2}]$ .

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

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

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

Multiply $2$ by $- 1$ .

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

Combine $- 2$ and $\frac{5}{2}$ .

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

Multiply $- 2$ by $5$ .

$x^{2} + \frac{- 10}{2} x + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 3]$

Combine $\frac{- 10}{2}$ and $x$ .

$x^{2} + \frac{- 10 x}{2} + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 3]$

Cancel the common factor of $- 10$ and $2$ .

Factor $2$ out of $- 10 x$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

$x^{2} + \frac{2 (- 5 x)}{2 (1)} + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 3]$

Cancel the common factor.

$x^{2} + \frac{2 (- 5 x)}{2 \cdot 1} + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 3]$

Rewrite the expression.

$x^{2} + \frac{- 5 x}{1} + \frac{d}{d x} [6 x] + \frac{d}{d x} [- 3]$

Divide $- 5 x$ by $1$ .

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

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

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

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

$x^{2} - 5 x + 6 \cdot 1 + \frac{d}{d x} [- 3]$

Multiply $6$ by $1$ .

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

Differentiate using the Constant Rule.

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

$x^{2} - 5 x + 6 + 0$

Add $x^{2} - 5 x + 6$ and $0$ .

$x^{2} - 5 x + 6$

Step 3

Find the second derivative of the function.

Differentiate.

By the Sum Rule, the derivative of $x^{2} - 5 x + 6$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [6]$ .

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

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

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

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

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

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

$f'' (x) = 2 x - 5 \cdot 1 + \frac{d}{d x} (6)$

Multiply $- 5$ by $1$ .

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

Differentiate using the Constant Rule.

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$f'' (x) = 2 x - 5 + 0$

Add $2 x - 5$ and $0$ .

$f'' (x) = 2 x - 5$

Step 4

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

$x^{2} - 5 x + 6 = 0$

Step 5

Find the first derivative.

$f' (x) = \frac{d}{d x} (\frac{1}{3} x^{3}) + \frac{d}{d x} (- \frac{5}{2} x^{2}) + \frac{d}{d x} (6 x) + \frac{d}{d x} (- 3)$

Evaluate $\frac{d}{d x} [\frac{1}{3} x^{3}]$ .

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{1}{3} x^{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x^{3}]$ .

$f' (x) = \frac{1}{3} \cdot \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- \frac{5}{2} x^{2}) + \frac{d}{d x} (6 x) + \frac{d}{d x} (- 3)$

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

$f' (x) = \frac{1}{3} \cdot (3 x^{2}) + \frac{d}{d x} (- \frac{5}{2} x^{2}) + \frac{d}{d x} (6 x) + \frac{d}{d x} (- 3)$

Combine $3$ and $\frac{1}{3}$ .

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

Combine $\frac{3}{3}$ and $x^{2}$ .

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

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

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

Evaluate $\frac{d}{d x} [- \frac{5}{2} x^{2}]$ .

Since $- \frac{5}{2}$ is constant with respect to $x$ , the derivative of $- \frac{5}{2} x^{2}$ with respect to $x$ is $- \frac{5}{2} \frac{d}{d x} [x^{2}]$ .

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

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

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

Multiply $2$ by $- 1$ .

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

Combine $- 2$ and $\frac{5}{2}$ .

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

Multiply $- 2$ by $5$ .

$f' (x) = x^{2} + \frac{- 10}{2} x + \frac{d}{d x} (6 x) + \frac{d}{d x} (- 3)$

Combine $\frac{- 10}{2}$ and $x$ .

$f' (x) = x^{2} + \frac{- 10 x}{2} + \frac{d}{d x} (6 x) + \frac{d}{d x} (- 3)$

Cancel the common factor of $- 10$ and $2$ .

Factor $2$ out of $- 10 x$ .

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

Cancel the common factors.

Factor $2$ out of $2$ .

$f' (x) = x^{2} + \frac{2 (- 5 x)}{2 (1)} + \frac{d}{d x} (6 x) + \frac{d}{d x} (- 3)$

Cancel the common factor.

$f' (x) = x^{2} + \frac{2 (- 5 x)}{2 \cdot 1} + \frac{d}{d x} (6 x) + \frac{d}{d x} (- 3)$

Rewrite the expression.

$f' (x) = x^{2} + \frac{- 5 x}{1} + \frac{d}{d x} (6 x) + \frac{d}{d x} (- 3)$

Divide $- 5 x$ by $1$ .

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

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

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

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

$f' (x) = x^{2} - 5 x + 6 \cdot 1 + \frac{d}{d x} (- 3)$

Multiply $6$ by $1$ .

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

Differentiate using the Constant Rule.

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

$f' (x) = x^{2} - 5 x + 6 + 0$

Add $x^{2} - 5 x + 6$ and $0$ .

$f' (x) = x^{2} - 5 x + 6$

The first derivative of $f (x)$ with respect to $x$ is $x^{2} - 5 x + 6$ .

$x^{2} - 5 x + 6$

Step 6

Set the first derivative equal to $0$ then solve the equation $x^{2} - 5 x + 6 = 0$ .

Set the first derivative equal to $0$ .

$x^{2} - 5 x + 6 = 0$

Factor $x^{2} - 5 x + 6$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $6$ and whose sum is $- 5$ .

$- 3, - 2$

Write the factored form using these integers.

$(x - 3) (x - 2) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 3 = 0$

$x - 2 = 0$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Set $x - 2$ equal to $0$ and solve for $x$ .

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

The final solution is all the values that make $(x - 3) (x - 2) = 0$ true.

$x = 3, 2$

Step 7

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.

Step 8

Critical points to evaluate.

$x = 3, 2$

Step 9

Evaluate the second derivative at $x = 3$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$2 (3) - 5$

Step 10

Evaluate the second derivative.

Multiply $2$ by $3$ .

$6 - 5$

Subtract $5$ from $6$ .

$1$

Step 11

$x = 3$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 3$ is a local minimum

Step 12

Find the y-value when $x = 3$ .

Replace the variable $x$ with $3$ in the expression.

$f (3) = \frac{1}{3} \cdot {(3)}^{3} - \frac{5}{2} \cdot {(3)}^{2} + 6 (3) - 3$

Simplify the result.

Simplify each term.

Cancel the common factor of $3$ .

Factor $3$ out of ${(3)}^{3}$ .

$f (3) = \frac{1}{3} \cdot (3 \cdot 3^{2}) - \frac{5}{2} \cdot {(3)}^{2} + 6 (3) - 3$

Cancel the common factor.

$f (3) = \frac{1}{3} \cdot (3 \cdot 3^{2}) - \frac{5}{2} \cdot {(3)}^{2} + 6 (3) - 3$

Rewrite the expression.

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

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

$f (3) = 9 - \frac{5}{2} \cdot {(3)}^{2} + 6 (3) - 3$

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

$f (3) = 9 - \frac{5}{2} \cdot 9 + 6 (3) - 3$

Multiply $- \frac{5}{2} \cdot 9$ .

Multiply $9$ by $- 1$ .

$f (3) = 9 - 9 (\frac{5}{2}) + 6 (3) - 3$

Combine $- 9$ and $\frac{5}{2}$ .

$f (3) = 9 + \frac{- 9 \cdot 5}{2} + 6 (3) - 3$

Multiply $- 9$ by $5$ .

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

Move the negative in front of the fraction.

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

Multiply $6$ by $3$ .

$f (3) = 9 - \frac{45}{2} + 18 - 3$

Find the common denominator.

Write $9$ as a fraction with denominator $1$ .

$f (3) = \frac{9}{1} - \frac{45}{2} + 18 - 3$

Multiply $\frac{9}{1}$ by $\frac{2}{2}$ .

$f (3) = \frac{9}{1} \cdot \frac{2}{2} - \frac{45}{2} + 18 - 3$

Multiply $\frac{9}{1}$ by $\frac{2}{2}$ .

$f (3) = \frac{9 \cdot 2}{2} - \frac{45}{2} + 18 - 3$

Write $18$ as a fraction with denominator $1$ .

$f (3) = \frac{9 \cdot 2}{2} - \frac{45}{2} + \frac{18}{1} - 3$

Multiply $\frac{18}{1}$ by $\frac{2}{2}$ .

$f (3) = \frac{9 \cdot 2}{2} - \frac{45}{2} + \frac{18}{1} \cdot \frac{2}{2} - 3$

Multiply $\frac{18}{1}$ by $\frac{2}{2}$ .

$f (3) = \frac{9 \cdot 2}{2} - \frac{45}{2} + \frac{18 \cdot 2}{2} - 3$

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

$f (3) = \frac{9 \cdot 2}{2} - \frac{45}{2} + \frac{18 \cdot 2}{2} + \frac{- 3}{1}$

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

$f (3) = \frac{9 \cdot 2}{2} - \frac{45}{2} + \frac{18 \cdot 2}{2} + \frac{- 3}{1} \cdot \frac{2}{2}$

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

$f (3) = \frac{9 \cdot 2}{2} - \frac{45}{2} + \frac{18 \cdot 2}{2} + \frac{- 3 \cdot 2}{2}$

Combine the numerators over the common denominator.

$f (3) = \frac{9 \cdot 2 - 45 + 18 \cdot 2 - 3 \cdot 2}{2}$

Simplify each term.

Multiply $9$ by $2$ .

$f (3) = \frac{18 - 45 + 18 \cdot 2 - 3 \cdot 2}{2}$

Multiply $18$ by $2$ .

$f (3) = \frac{18 - 45 + 36 - 3 \cdot 2}{2}$

Multiply $- 3$ by $2$ .

$f (3) = \frac{18 - 45 + 36 - 6}{2}$

Simplify by adding and subtracting.

Subtract $45$ from $18$ .

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

Add $- 27$ and $36$ .

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

Subtract $6$ from $9$ .

$f (3) = \frac{3}{2}$

The final answer is $\frac{3}{2}$ .

$y = \frac{3}{2}$

Step 13

Evaluate the second derivative at $x = 2$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$2 (2) - 5$

Step 14

Evaluate the second derivative.

Multiply $2$ by $2$ .

$4 - 5$

Subtract $5$ from $4$ .

$- 1$

Step 15

$x = 2$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = 2$ is a local maximum

Step 16

Find the y-value when $x = 2$ .

Replace the variable $x$ with $2$ in the expression.

$f (2) = \frac{1}{3} \cdot {(2)}^{3} - \frac{5}{2} \cdot {(2)}^{2} + 6 (2) - 3$

Simplify the result.

Simplify each term.

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

$f (2) = \frac{1}{3} \cdot 8 - \frac{5}{2} \cdot {(2)}^{2} + 6 (2) - 3$

Combine $\frac{1}{3}$ and $8$ .

$f (2) = \frac{8}{3} - \frac{5}{2} \cdot {(2)}^{2} + 6 (2) - 3$

Cancel the common factor of $2$ .

Move the leading negative in $- \frac{5}{2}$ into the numerator.

$f (2) = \frac{8}{3} + \frac{- 5}{2} \cdot {(2)}^{2} + 6 (2) - 3$

Factor $2$ out of ${(2)}^{2}$ .

$f (2) = \frac{8}{3} + \frac{- 5}{2} \cdot (2 \cdot 2) + 6 (2) - 3$

Cancel the common factor.

$f (2) = \frac{8}{3} + \frac{- 5}{2} \cdot (2 \cdot 2) + 6 (2) - 3$

Rewrite the expression.

$f (2) = \frac{8}{3} - 5 \cdot 2 + 6 (2) - 3$

Multiply $- 5$ by $2$ .

$f (2) = \frac{8}{3} - 10 + 6 (2) - 3$

Multiply $6$ by $2$ .

$f (2) = \frac{8}{3} - 10 + 12 - 3$

Find the common denominator.

Write $- 10$ as a fraction with denominator $1$ .

$f (2) = \frac{8}{3} + \frac{- 10}{1} + 12 - 3$

Multiply $\frac{- 10}{1}$ by $\frac{3}{3}$ .

$f (2) = \frac{8}{3} + \frac{- 10}{1} \cdot \frac{3}{3} + 12 - 3$

Multiply $\frac{- 10}{1}$ by $\frac{3}{3}$ .

$f (2) = \frac{8}{3} + \frac{- 10 \cdot 3}{3} + 12 - 3$

Write $12$ as a fraction with denominator $1$ .

$f (2) = \frac{8}{3} + \frac{- 10 \cdot 3}{3} + \frac{12}{1} - 3$

Multiply $\frac{12}{1}$ by $\frac{3}{3}$ .

$f (2) = \frac{8}{3} + \frac{- 10 \cdot 3}{3} + \frac{12}{1} \cdot \frac{3}{3} - 3$

Multiply $\frac{12}{1}$ by $\frac{3}{3}$ .

$f (2) = \frac{8}{3} + \frac{- 10 \cdot 3}{3} + \frac{12 \cdot 3}{3} - 3$

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

$f (2) = \frac{8}{3} + \frac{- 10 \cdot 3}{3} + \frac{12 \cdot 3}{3} + \frac{- 3}{1}$

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

$f (2) = \frac{8}{3} + \frac{- 10 \cdot 3}{3} + \frac{12 \cdot 3}{3} + \frac{- 3}{1} \cdot \frac{3}{3}$

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

$f (2) = \frac{8}{3} + \frac{- 10 \cdot 3}{3} + \frac{12 \cdot 3}{3} + \frac{- 3 \cdot 3}{3}$

Combine the numerators over the common denominator.

$f (2) = \frac{8 - 10 \cdot 3 + 12 \cdot 3 - 3 \cdot 3}{3}$

Simplify each term.

Multiply $- 10$ by $3$ .

$f (2) = \frac{8 - 30 + 12 \cdot 3 - 3 \cdot 3}{3}$

Multiply $12$ by $3$ .

$f (2) = \frac{8 - 30 + 36 - 3 \cdot 3}{3}$

Multiply $- 3$ by $3$ .

$f (2) = \frac{8 - 30 + 36 - 9}{3}$

Simplify by adding and subtracting.

Subtract $30$ from $8$ .

$f (2) = \frac{- 22 + 36 - 9}{3}$

Add $- 22$ and $36$ .

$f (2) = \frac{14 - 9}{3}$

Subtract $9$ from $14$ .

$f (2) = \frac{5}{3}$

The final answer is $\frac{5}{3}$ .

$y = \frac{5}{3}$

Step 17

These are the local extrema for $f (x) = \frac{1}{3} x^{3} - \frac{5}{2} x^{2} + 6 x - 3$ .

$(3, \frac{3}{2})$ is a local minima

$(2, \frac{5}{3})$ is a local maxima

Step 18

Find the Local Maxima and Minima y=1/3x^3-5/2x^2+6x-3

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