Calculus Examples

$P (x) = \frac{1}{9} x^{4} - \frac{4}{9} x^{3}$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $\frac{1}{9} x^{4} - \frac{4}{9} x^{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{9} x^{4}] + \frac{d}{d x} [- \frac{4}{9} x^{3}]$ .

$\frac{d}{d x} [\frac{1}{9} x^{4}] + \frac{d}{d x} [- \frac{4}{9} x^{3}]$

Step 1.2

Evaluate $\frac{d}{d x} [\frac{1}{9} x^{4}]$ .

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Step 1.2.1

Since $\frac{1}{9}$ is constant with respect to $x$ , the derivative of $\frac{1}{9} x^{4}$ with respect to $x$ is $\frac{1}{9} \frac{d}{d x} [x^{4}]$ .

$\frac{1}{9} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{4}{9} x^{3}]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{1}{9} (4 x^{3}) + \frac{d}{d x} [- \frac{4}{9} x^{3}]$

Step 1.2.3

Combine $4$ and $\frac{1}{9}$ .

$\frac{4}{9} x^{3} + \frac{d}{d x} [- \frac{4}{9} x^{3}]$

Step 1.2.4

Combine $\frac{4}{9}$ and $x^{3}$ .

$\frac{4 x^{3}}{9} + \frac{d}{d x} [- \frac{4}{9} x^{3}]$

Step 1.3

Evaluate $\frac{d}{d x} [- \frac{4}{9} x^{3}]$ .

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Step 1.3.1

Since $- \frac{4}{9}$ is constant with respect to $x$ , the derivative of $- \frac{4}{9} x^{3}$ with respect to $x$ is $- \frac{4}{9} \frac{d}{d x} [x^{3}]$ .

$\frac{4 x^{3}}{9} - \frac{4}{9} \frac{d}{d x} [x^{3}]$

Step 1.3.2

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

$\frac{4 x^{3}}{9} - \frac{4}{9} (3 x^{2})$

Step 1.3.3

Multiply $3$ by $- 1$ .

$\frac{4 x^{3}}{9} - 3 (\frac{4}{9}) x^{2}$

Step 1.3.4

Combine $- 3$ and $\frac{4}{9}$ .

$\frac{4 x^{3}}{9} + \frac{- 3 \cdot 4}{9} x^{2}$

Step 1.3.5

Multiply $- 3$ by $4$ .

$\frac{4 x^{3}}{9} + \frac{- 12}{9} x^{2}$

Step 1.3.6

Combine $\frac{- 12}{9}$ and $x^{2}$ .

$\frac{4 x^{3}}{9} + \frac{- 12 x^{2}}{9}$

Step 1.3.7

Cancel the common factor of $- 12$ and $9$ .

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Step 1.3.7.1

Factor $3$ out of $- 12 x^{2}$ .

$\frac{4 x^{3}}{9} + \frac{3 (- 4 x^{2})}{9}$

Step 1.3.7.2

Cancel the common factors.

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Step 1.3.7.2.1

Factor $3$ out of $9$ .

$\frac{4 x^{3}}{9} + \frac{3 (- 4 x^{2})}{3 (3)}$

Step 1.3.7.2.2

Cancel the common factor.

$\frac{4 x^{3}}{9} + \frac{3 (- 4 x^{2})}{3 \cdot 3}$

Step 1.3.7.2.3

Rewrite the expression.

$\frac{4 x^{3}}{9} + \frac{- 4 x^{2}}{3}$

Step 1.3.8

Move the negative in front of the fraction.

$\frac{4 x^{3}}{9} - \frac{4 x^{2}}{3}$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $\frac{4 x^{3}}{9} - \frac{4 x^{2}}{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{4 x^{3}}{9}] + \frac{d}{d x} [- \frac{4 x^{2}}{3}]$ .

$P'' (x) = \frac{d}{d x} (\frac{4 x^{3}}{9}) + \frac{d}{d x} (- \frac{4 x^{2}}{3})$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{4 x^{3}}{9}]$ .

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Step 2.2.1

Since $\frac{4}{9}$ is constant with respect to $x$ , the derivative of $\frac{4 x^{3}}{9}$ with respect to $x$ is $\frac{4}{9} \frac{d}{d x} [x^{3}]$ .

$P'' (x) = \frac{4}{9} \cdot \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- \frac{4 x^{2}}{3})$

Step 2.2.2

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

$P'' (x) = \frac{4}{9} \cdot (3 x^{2}) + \frac{d}{d x} (- \frac{4 x^{2}}{3})$

Step 2.2.3

Combine $3$ and $\frac{4}{9}$ .

$P'' (x) = \frac{3 \cdot 4}{9} x^{2} + \frac{d}{d x} (- \frac{4 x^{2}}{3})$

Step 2.2.4

Multiply $3$ by $4$ .

$P'' (x) = \frac{12}{9} x^{2} + \frac{d}{d x} (- \frac{4 x^{2}}{3})$

Step 2.2.5

Combine $\frac{12}{9}$ and $x^{2}$ .

$P'' (x) = \frac{12 x^{2}}{9} + \frac{d}{d x} (- \frac{4 x^{2}}{3})$

Step 2.2.6

Cancel the common factor of $12$ and $9$ .

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Step 2.2.6.1

Factor $3$ out of $12 x^{2}$ .

$P'' (x) = \frac{3 (4 x^{2})}{9} + \frac{d}{d x} (- \frac{4 x^{2}}{3})$

Step 2.2.6.2

Cancel the common factors.

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Step 2.2.6.2.1

Factor $3$ out of $9$ .

$P'' (x) = \frac{3 (4 x^{2})}{3 (3)} + \frac{d}{d x} (- \frac{4 x^{2}}{3})$

Step 2.2.6.2.2

Cancel the common factor.

$P'' (x) = \frac{3 (4 x^{2})}{3 \cdot 3} + \frac{d}{d x} (- \frac{4 x^{2}}{3})$

Step 2.2.6.2.3

Rewrite the expression.

$P'' (x) = \frac{4 x^{2}}{3} + \frac{d}{d x} (- \frac{4 x^{2}}{3})$

Step 2.3

Evaluate $\frac{d}{d x} [- \frac{4 x^{2}}{3}]$ .

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Step 2.3.1

Since $- \frac{4}{3}$ is constant with respect to $x$ , the derivative of $- \frac{4 x^{2}}{3}$ with respect to $x$ is $- \frac{4}{3} \frac{d}{d x} [x^{2}]$ .

$P'' (x) = \frac{4 x^{2}}{3} - \frac{4}{3} \cdot \frac{d}{d x} x^{2}$

Step 2.3.2

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

$P'' (x) = \frac{4 x^{2}}{3} - \frac{4}{3} \cdot (2 x)$

Step 2.3.3

Multiply $2$ by $- 1$ .

$P'' (x) = \frac{4 x^{2}}{3} - 2 (\frac{4}{3}) \cdot x$

Step 2.3.4

Combine $- 2$ and $\frac{4}{3}$ .

$P'' (x) = \frac{4 x^{2}}{3} + \frac{- 2 \cdot 4}{3} x$

Step 2.3.5

Multiply $- 2$ by $4$ .

$P'' (x) = \frac{4 x^{2}}{3} + \frac{- 8}{3} x$

Step 2.3.6

Combine $\frac{- 8}{3}$ and $x$ .

$P'' (x) = \frac{4 x^{2}}{3} + \frac{- 8 x}{3}$

Step 2.3.7

Move the negative in front of the fraction.

$P'' (x) = \frac{4 x^{2}}{3} - \frac{8 x}{3}$

Step 3

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

$\frac{4 x^{3}}{9} - \frac{4 x^{2}}{3} = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

By the Sum Rule, the derivative of $\frac{1}{9} x^{4} - \frac{4}{9} x^{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{9} x^{4}] + \frac{d}{d x} [- \frac{4}{9} x^{3}]$ .

$\frac{d}{d x} [\frac{1}{9} x^{4}] + \frac{d}{d x} [- \frac{4}{9} x^{3}]$

Step 4.1.2

Evaluate $\frac{d}{d x} [\frac{1}{9} x^{4}]$ .

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Step 4.1.2.1

Since $\frac{1}{9}$ is constant with respect to $x$ , the derivative of $\frac{1}{9} x^{4}$ with respect to $x$ is $\frac{1}{9} \frac{d}{d x} [x^{4}]$ .

$\frac{1}{9} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{4}{9} x^{3}]$

Step 4.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{1}{9} (4 x^{3}) + \frac{d}{d x} [- \frac{4}{9} x^{3}]$

Step 4.1.2.3

Combine $4$ and $\frac{1}{9}$ .

$\frac{4}{9} x^{3} + \frac{d}{d x} [- \frac{4}{9} x^{3}]$

Step 4.1.2.4

Combine $\frac{4}{9}$ and $x^{3}$ .

$\frac{4 x^{3}}{9} + \frac{d}{d x} [- \frac{4}{9} x^{3}]$

Step 4.1.3

Evaluate $\frac{d}{d x} [- \frac{4}{9} x^{3}]$ .

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Step 4.1.3.1

Since $- \frac{4}{9}$ is constant with respect to $x$ , the derivative of $- \frac{4}{9} x^{3}$ with respect to $x$ is $- \frac{4}{9} \frac{d}{d x} [x^{3}]$ .

$\frac{4 x^{3}}{9} - \frac{4}{9} \frac{d}{d x} [x^{3}]$

Step 4.1.3.2

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

$\frac{4 x^{3}}{9} - \frac{4}{9} (3 x^{2})$

Step 4.1.3.3

Multiply $3$ by $- 1$ .

$\frac{4 x^{3}}{9} - 3 (\frac{4}{9}) x^{2}$

Step 4.1.3.4

Combine $- 3$ and $\frac{4}{9}$ .

$\frac{4 x^{3}}{9} + \frac{- 3 \cdot 4}{9} x^{2}$

Step 4.1.3.5

Multiply $- 3$ by $4$ .

$\frac{4 x^{3}}{9} + \frac{- 12}{9} x^{2}$

Step 4.1.3.6

Combine $\frac{- 12}{9}$ and $x^{2}$ .

$\frac{4 x^{3}}{9} + \frac{- 12 x^{2}}{9}$

Step 4.1.3.7

Cancel the common factor of $- 12$ and $9$ .

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Step 4.1.3.7.1

Factor $3$ out of $- 12 x^{2}$ .

$\frac{4 x^{3}}{9} + \frac{3 (- 4 x^{2})}{9}$

Step 4.1.3.7.2

Cancel the common factors.

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Step 4.1.3.7.2.1

Factor $3$ out of $9$ .

$\frac{4 x^{3}}{9} + \frac{3 (- 4 x^{2})}{3 (3)}$

Step 4.1.3.7.2.2

Cancel the common factor.

$\frac{4 x^{3}}{9} + \frac{3 (- 4 x^{2})}{3 \cdot 3}$

Step 4.1.3.7.2.3

Rewrite the expression.

$\frac{4 x^{3}}{9} + \frac{- 4 x^{2}}{3}$

Step 4.1.3.8

Move the negative in front of the fraction.

$f' (x) = \frac{4 x^{3}}{9} - \frac{4 x^{2}}{3}$

Step 4.2

The first derivative of $P (x)$ with respect to $x$ is $\frac{4 x^{3}}{9} - \frac{4 x^{2}}{3}$ .

$\frac{4 x^{3}}{9} - \frac{4 x^{2}}{3}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{4 x^{3}}{9} - \frac{4 x^{2}}{3} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{4 x^{3}}{9} - \frac{4 x^{2}}{3} = 0$

Step 5.2

Multiply each term in $\frac{4 x^{3}}{9} - \frac{4 x^{2}}{3} = 0$ by $9$ to eliminate the fractions.

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Step 5.2.1

Multiply each term in $\frac{4 x^{3}}{9} - \frac{4 x^{2}}{3} = 0$ by $9$ .

$\frac{4 x^{3}}{9} \cdot 9 - \frac{4 x^{2}}{3} \cdot 9 = 0 \cdot 9$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Simplify each term.

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Step 5.2.2.1.1

Cancel the common factor of $9$ .

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Step 5.2.2.1.1.1

Cancel the common factor.

$\frac{4 x^{3}}{9} \cdot 9 - \frac{4 x^{2}}{3} \cdot 9 = 0 \cdot 9$

Step 5.2.2.1.1.2

Rewrite the expression.

$4 x^{3} - \frac{4 x^{2}}{3} \cdot 9 = 0 \cdot 9$

Step 5.2.2.1.2

Cancel the common factor of $3$ .

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Step 5.2.2.1.2.1

Move the leading negative in $- \frac{4 x^{2}}{3}$ into the numerator.

$4 x^{3} + \frac{- 4 x^{2}}{3} \cdot 9 = 0 \cdot 9$

Step 5.2.2.1.2.2

Factor $3$ out of $9$ .

$4 x^{3} + \frac{- 4 x^{2}}{3} \cdot (3 (3)) = 0 \cdot 9$

Step 5.2.2.1.2.3

Cancel the common factor.

$4 x^{3} + \frac{- 4 x^{2}}{3} \cdot (3 \cdot 3) = 0 \cdot 9$

Step 5.2.2.1.2.4

Rewrite the expression.

$4 x^{3} - 4 x^{2} \cdot 3 = 0 \cdot 9$

Step 5.2.2.1.3

Multiply $3$ by $- 4$ .

$4 x^{3} - 12 x^{2} = 0 \cdot 9$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Multiply $0$ by $9$ .

$4 x^{3} - 12 x^{2} = 0$

Step 5.3

Factor $4 x^{2}$ out of $4 x^{3} - 12 x^{2}$ .

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Step 5.3.1

Factor $4 x^{2}$ out of $4 x^{3}$ .

$4 x^{2} (x) - 12 x^{2} = 0$

Step 5.3.2

Factor $4 x^{2}$ out of $- 12 x^{2}$ .

$4 x^{2} (x) + 4 x^{2} (- 3) = 0$

Step 5.3.3

Factor $4 x^{2}$ out of $4 x^{2} (x) + 4 x^{2} (- 3)$ .

$4 x^{2} (x - 3) = 0$

Step 5.4

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

$x^{2} = 0$

$x - 3 = 0$

Step 5.5

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 5.5.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 5.5.2

Solve $x^{2} = 0$ for $x$ .

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Step 5.5.2.1

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

$x = \pm \sqrt{0}$

Step 5.5.2.2

Simplify $\pm \sqrt{0}$ .

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Step 5.5.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 5.5.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 5.5.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5.6

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

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Step 5.6.1

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

$x - 3 = 0$

Step 5.6.2

Add $3$ to both sides of the equation.

$x = 3$

Step 5.7

The final solution is all the values that make $4 x^{2} (x - 3) = 0$ true.

$x = 0, 3$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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 7

Critical points to evaluate.

$x = 0, 3$

Step 8

Evaluate the second derivative at $x = 0$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{4 {(0)}^{2}}{3} - \frac{8 (0)}{3}$

Step 9

Evaluate the second derivative.

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Step 9.1

Combine the numerators over the common denominator.

$\frac{4 {(0)}^{2} - 8 \cdot 0}{3}$

Step 9.2

Simplify each term.

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Step 9.2.1

Raising $0$ to any positive power yields $0$ .

$\frac{4 \cdot 0 - 8 \cdot 0}{3}$

Step 9.2.2

Multiply $4$ by $0$ .

$\frac{0 - 8 \cdot 0}{3}$

Step 9.2.3

Multiply $- 8$ by $0$ .

$\frac{0 + 0}{3}$

Step 9.3

Simplify the expression.

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Step 9.3.1

Add $0$ and $0$ .

$\frac{0}{3}$

Step 9.3.2

Divide $0$ by $3$ .

$0$

Step 10

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 10.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, 0) \cup (0, 3) \cup (3, \infty)$

Step 10.2

Substitute any number, such as $- 2$ , from the interval $(- \infty, 0)$ in the first derivative $\frac{4 x^{3}}{9} - \frac{4 x^{2}}{3}$ to check if the result is negative or positive.

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Step 10.2.1

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

$P' (- 2) = \frac{4 {(- 2)}^{3}}{9} - \frac{4 {(- 2)}^{2}}{3}$

Step 10.2.2

Simplify the result.

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Step 10.2.2.1

Simplify each term.

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Step 10.2.2.1.1

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

$P' (- 2) = \frac{4 \cdot - 8}{9} - \frac{4 {(- 2)}^{2}}{3}$

Step 10.2.2.1.2

Multiply $4$ by $- 8$ .

$P' (- 2) = \frac{- 32}{9} - \frac{4 {(- 2)}^{2}}{3}$

Step 10.2.2.1.3

Move the negative in front of the fraction.

$P' (- 2) = - \frac{32}{9} - \frac{4 {(- 2)}^{2}}{3}$

Step 10.2.2.1.4

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

$P' (- 2) = - \frac{32}{9} - \frac{4 \cdot 4}{3}$

Step 10.2.2.1.5

Multiply $4$ by $4$ .

$P' (- 2) = - \frac{32}{9} - \frac{16}{3}$

Step 10.2.2.2

To write $- \frac{16}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$P' (- 2) = - \frac{32}{9} - \frac{16}{3} \cdot \frac{3}{3}$

Step 10.2.2.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.2.2.3.1

Multiply $\frac{16}{3}$ by $\frac{3}{3}$ .

$P' (- 2) = - \frac{32}{9} - \frac{16 \cdot 3}{3 \cdot 3}$

Step 10.2.2.3.2

Multiply $3$ by $3$ .

$P' (- 2) = - \frac{32}{9} - \frac{16 \cdot 3}{9}$

Step 10.2.2.4

Combine the numerators over the common denominator.

$P' (- 2) = \frac{- 32 - 16 \cdot 3}{9}$

Step 10.2.2.5

Simplify the numerator.

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Step 10.2.2.5.1

Multiply $- 16$ by $3$ .

$P' (- 2) = \frac{- 32 - 48}{9}$

Step 10.2.2.5.2

Subtract $48$ from $- 32$ .

$P' (- 2) = \frac{- 80}{9}$

Step 10.2.2.6

Move the negative in front of the fraction.

$P' (- 2) = - \frac{80}{9}$

Step 10.2.2.7

The final answer is $- \frac{80}{9}$ .

$- \frac{80}{9}$

Step 10.3

Substitute any number, such as $2$ , from the interval $(0, 3)$ in the first derivative $\frac{4 x^{3}}{9} - \frac{4 x^{2}}{3}$ to check if the result is negative or positive.

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Step 10.3.1

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

$P' (2) = \frac{4 {(2)}^{3}}{9} - \frac{4 {(2)}^{2}}{3}$

Step 10.3.2

Simplify the result.

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Step 10.3.2.1

Simplify each term.

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Step 10.3.2.1.1

Simplify the numerator.

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Step 10.3.2.1.1.1

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

$P' (2) = \frac{2^{2} \cdot 2^{3}}{9} - \frac{4 {(2)}^{2}}{3}$

Step 10.3.2.1.1.2

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

$P' (2) = \frac{2^{2 + 3}}{9} - \frac{4 {(2)}^{2}}{3}$

Step 10.3.2.1.1.3

Add $2$ and $3$ .

$P' (2) = \frac{2^{5}}{9} - \frac{4 {(2)}^{2}}{3}$

Step 10.3.2.1.2

Raise $2$ to the power of $5$ .

$P' (2) = \frac{32}{9} - \frac{4 {(2)}^{2}}{3}$

Step 10.3.2.1.3

Simplify the numerator.

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Step 10.3.2.1.3.1

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

$P' (2) = \frac{32}{9} - \frac{2^{2} \cdot 2^{2}}{3}$

Step 10.3.2.1.3.2

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

$P' (2) = \frac{32}{9} - \frac{2^{2 + 2}}{3}$

Step 10.3.2.1.3.3

Add $2$ and $2$ .

$P' (2) = \frac{32}{9} - \frac{2^{4}}{3}$

Step 10.3.2.1.4

Raise $2$ to the power of $4$ .

$P' (2) = \frac{32}{9} - \frac{16}{3}$

Step 10.3.2.2

To write $- \frac{16}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$P' (2) = \frac{32}{9} - \frac{16}{3} \cdot \frac{3}{3}$

Step 10.3.2.3

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.3.2.3.1

Multiply $\frac{16}{3}$ by $\frac{3}{3}$ .

$P' (2) = \frac{32}{9} - \frac{16 \cdot 3}{3 \cdot 3}$

Step 10.3.2.3.2

Multiply $3$ by $3$ .

$P' (2) = \frac{32}{9} - \frac{16 \cdot 3}{9}$

Step 10.3.2.4

Combine the numerators over the common denominator.

$P' (2) = \frac{32 - 16 \cdot 3}{9}$

Step 10.3.2.5

Simplify the numerator.

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Step 10.3.2.5.1

Multiply $- 16$ by $3$ .

$P' (2) = \frac{32 - 48}{9}$

Step 10.3.2.5.2

Subtract $48$ from $32$ .

$P' (2) = \frac{- 16}{9}$

Step 10.3.2.6

Move the negative in front of the fraction.

$P' (2) = - \frac{16}{9}$

Step 10.3.2.7

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

$- \frac{16}{9}$

Step 10.4

Substitute any number, such as $6$ , from the interval $(3, \infty)$ in the first derivative $\frac{4 x^{3}}{9} - \frac{4 x^{2}}{3}$ to check if the result is negative or positive.

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Step 10.4.1

Replace the variable $x$ with $6$ in the expression.

$P' (6) = \frac{4 {(6)}^{3}}{9} - \frac{4 {(6)}^{2}}{3}$

Step 10.4.2

Simplify the result.

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Step 10.4.2.1

Simplify each term.

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Step 10.4.2.1.1

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

$P' (6) = \frac{4 \cdot 216}{9} - \frac{4 {(6)}^{2}}{3}$

Step 10.4.2.1.2

Multiply $4$ by $216$ .

$P' (6) = \frac{864}{9} - \frac{4 {(6)}^{2}}{3}$

Step 10.4.2.1.3

Divide $864$ by $9$ .

$P' (6) = 96 - \frac{4 {(6)}^{2}}{3}$

Step 10.4.2.1.4

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

$P' (6) = 96 - \frac{4 \cdot 36}{3}$

Step 10.4.2.1.5

Multiply $4$ by $36$ .

$P' (6) = 96 - \frac{144}{3}$

Step 10.4.2.1.6

Divide $144$ by $3$ .

$P' (6) = 96 - 1 \cdot 48$

Step 10.4.2.1.7

Multiply $- 1$ by $48$ .

$P' (6) = 96 - 48$

Step 10.4.2.2

Subtract $48$ from $96$ .

$P' (6) = 48$

Step 10.4.2.3

The final answer is $48$ .

$48$

Step 10.5

Since the first derivative did not change signs around $x = 0$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 10.6

Since the first derivative changed signs from negative to positive around $x = 3$ , then $x = 3$ is a local minimum.

$x = 3$ is a local minimum

Step 11