Calculus Examples

$f (x) = 9 x^{2} - \frac{3}{2} x^{3}$ , $[0, 6]$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

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

$\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [- \frac{3}{2} x^{3}]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [9 x^{2}]$ .

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Step 1.1.1.2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \frac{3}{2} x^{3}]$

Step 1.1.1.2.2

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

$9 (2 x) + \frac{d}{d x} [- \frac{3}{2} x^{3}]$

Step 1.1.1.2.3

Multiply $2$ by $9$ .

$18 x + \frac{d}{d x} [- \frac{3}{2} x^{3}]$

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

$18 x - \frac{3}{2} \frac{d}{d x} [x^{3}]$

Step 1.1.1.3.2

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

$18 x - \frac{3}{2} (3 x^{2})$

Step 1.1.1.3.3

Multiply $3$ by $- 1$ .

$18 x - 3 (\frac{3}{2}) x^{2}$

Step 1.1.1.3.4

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

$18 x + \frac{- 3 \cdot 3}{2} x^{2}$

Step 1.1.1.3.5

Multiply $- 3$ by $3$ .

$18 x + \frac{- 9}{2} x^{2}$

Step 1.1.1.3.6

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

$18 x + \frac{- 9 x^{2}}{2}$

Step 1.1.1.3.7

Move the negative in front of the fraction.

$18 x - \frac{9 x^{2}}{2}$

Step 1.1.1.4

Reorder terms.

$f' (x) = - \frac{9 x^{2}}{2} + 18 x$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{9 x^{2}}{2} + 18 x$ .

$- \frac{9 x^{2}}{2} + 18 x$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $- \frac{9 x^{2}}{2} + 18 x = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$- \frac{9 x^{2}}{2} + 18 x = 0$

Step 1.2.2

Multiply each term in $- \frac{9 x^{2}}{2} + 18 x = 0$ by $2$ to eliminate the fractions.

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Step 1.2.2.1

Multiply each term in $- \frac{9 x^{2}}{2} + 18 x = 0$ by $2$ .

$- \frac{9 x^{2}}{2} \cdot 2 + 18 x \cdot 2 = 0 \cdot 2$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Simplify each term.

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Step 1.2.2.2.1.1

Cancel the common factor of $2$ .

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Step 1.2.2.2.1.1.1

Move the leading negative in $- \frac{9 x^{2}}{2}$ into the numerator.

$\frac{- 9 x^{2}}{2} \cdot 2 + 18 x \cdot 2 = 0 \cdot 2$

Step 1.2.2.2.1.1.2

Cancel the common factor.

$\frac{- 9 x^{2}}{2} \cdot 2 + 18 x \cdot 2 = 0 \cdot 2$

Step 1.2.2.2.1.1.3

Rewrite the expression.

$- 9 x^{2} + 18 x \cdot 2 = 0 \cdot 2$

Step 1.2.2.2.1.2

Multiply $2$ by $18$ .

$- 9 x^{2} + 36 x = 0 \cdot 2$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Multiply $0$ by $2$ .

$- 9 x^{2} + 36 x = 0$

Step 1.2.3

Factor $- 9 x$ out of $- 9 x^{2} + 36 x$ .

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Step 1.2.3.1

Factor $- 9 x$ out of $- 9 x^{2}$ .

$- 9 x \cdot x + 36 x = 0$

Step 1.2.3.2

Factor $- 9 x$ out of $36 x$ .

$- 9 x \cdot x - 9 x \cdot - 4 = 0$

Step 1.2.3.3

Factor $- 9 x$ out of $- 9 x (x) - 9 x (- 4)$ .

$- 9 x (x - 4) = 0$

Step 1.2.4

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

$x = 0$

$x - 4 = 0$

Step 1.2.5

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.6

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 1.2.6.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 1.2.6.2

Add $4$ to both sides of the equation.

$x = 4$

Step 1.2.7

The final solution is all the values that make $- 9 x (x - 4) = 0$ true.

$x = 0, 4$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

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 1.4

Evaluate $9 x^{2} - \frac{3}{2} x^{3}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 0$ .

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Step 1.4.1.1

Substitute $0$ for $x$ .

$9 {(0)}^{2} - \frac{3}{2} \cdot {(0)}^{3}$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

Simplify each term.

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Step 1.4.1.2.1.1

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

$9 \cdot 0 - \frac{3}{2} \cdot {(0)}^{3}$

Step 1.4.1.2.1.2

Multiply $9$ by $0$ .

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

Step 1.4.1.2.1.3

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

$0 - \frac{3}{2} \cdot 0$

Step 1.4.1.2.1.4

Multiply $- \frac{3}{2} \cdot 0$ .

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Step 1.4.1.2.1.4.1

Multiply $0$ by $- 1$ .

$0 + 0 (\frac{3}{2})$

Step 1.4.1.2.1.4.2

Multiply $0$ by $\frac{3}{2}$ .

$0 + 0$

Step 1.4.1.2.2

Add $0$ and $0$ .

$0$

Step 1.4.2

Evaluate at $x = 4$ .

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Step 1.4.2.1

Substitute $4$ for $x$ .

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

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Simplify each term.

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Step 1.4.2.2.1.1

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

$9 \cdot 16 - \frac{3}{2} \cdot {(4)}^{3}$

Step 1.4.2.2.1.2

Multiply $9$ by $16$ .

$144 - \frac{3}{2} \cdot {(4)}^{3}$

Step 1.4.2.2.1.3

Raise $4$ to the power of $3$ .

$144 - \frac{3}{2} \cdot 64$

Step 1.4.2.2.1.4

Cancel the common factor of $2$ .

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Step 1.4.2.2.1.4.1

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

$144 + \frac{- 3}{2} \cdot 64$

Step 1.4.2.2.1.4.2

Factor $2$ out of $64$ .

$144 + \frac{- 3}{2} \cdot (2 (32))$

Step 1.4.2.2.1.4.3

Cancel the common factor.

$144 + \frac{- 3}{2} \cdot (2 \cdot 32)$

Step 1.4.2.2.1.4.4

Rewrite the expression.

$144 - 3 \cdot 32$

Step 1.4.2.2.1.5

Multiply $- 3$ by $32$ .

$144 - 96$

Step 1.4.2.2.2

Subtract $96$ from $144$ .

$48$

Step 1.4.3

List all of the points.

$(0, 0), (4, 48)$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = 0$ .

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Step 2.1.1

Substitute $0$ for $x$ .

$9 {(0)}^{2} - \frac{3}{2} \cdot {(0)}^{3}$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

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

$9 \cdot 0 - \frac{3}{2} \cdot {(0)}^{3}$

Step 2.1.2.1.2

Multiply $9$ by $0$ .

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

Step 2.1.2.1.3

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

$0 - \frac{3}{2} \cdot 0$

Step 2.1.2.1.4

Multiply $- \frac{3}{2} \cdot 0$ .

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Step 2.1.2.1.4.1

Multiply $0$ by $- 1$ .

$0 + 0 (\frac{3}{2})$

Step 2.1.2.1.4.2

Multiply $0$ by $\frac{3}{2}$ .

$0 + 0$

Step 2.1.2.2

Add $0$ and $0$ .

$0$

Step 2.2

Evaluate at $x = 6$ .

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Step 2.2.1

Substitute $6$ for $x$ .

$9 {(6)}^{2} - \frac{3}{2} \cdot {(6)}^{3}$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

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

$9 \cdot 36 - \frac{3}{2} \cdot {(6)}^{3}$

Step 2.2.2.1.2

Multiply $9$ by $36$ .

$324 - \frac{3}{2} \cdot {(6)}^{3}$

Step 2.2.2.1.3

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

$324 - \frac{3}{2} \cdot 216$

Step 2.2.2.1.4

Cancel the common factor of $2$ .

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Step 2.2.2.1.4.1

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

$324 + \frac{- 3}{2} \cdot 216$

Step 2.2.2.1.4.2

Factor $2$ out of $216$ .

$324 + \frac{- 3}{2} \cdot (2 (108))$

Step 2.2.2.1.4.3

Cancel the common factor.

$324 + \frac{- 3}{2} \cdot (2 \cdot 108)$

Step 2.2.2.1.4.4

Rewrite the expression.

$324 - 3 \cdot 108$

Step 2.2.2.1.5

Multiply $- 3$ by $108$ .

$324 - 324$

Step 2.2.2.2

Subtract $324$ from $324$ .

$0$

Step 2.3

List all of the points.

$(0, 0), (6, 0)$

Step 3

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

Absolute Maximum: $(4, 48)$

Absolute Minimum: $(0, 0), (6, 0)$

Step 4