Calculus Examples

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

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

Differentiate.

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Step 1.1.1.1.1

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

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

Step 1.1.1.1.2

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

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

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

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

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 $n = 2$ .

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

Step 1.1.1.2.3

Multiply $2$ by $- 1$ .

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

Step 1.1.1.2.4

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

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

Step 1.1.1.2.5

Multiply $- 2$ by $3$ .

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

Step 1.1.1.2.6

Combine $\frac{- 6}{2}$ and $x$ .

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

Step 1.1.1.2.7

Cancel the common factor of $- 6$ and $2$ .

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Step 1.1.1.2.7.1

Factor $2$ out of $- 6 x$ .

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

Step 1.1.1.2.7.2

Cancel the common factors.

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Step 1.1.1.2.7.2.1

Factor $2$ out of $2$ .

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

Step 1.1.1.2.7.2.2

Cancel the common factor.

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

Step 1.1.1.2.7.2.3

Rewrite the expression.

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

Step 1.1.1.2.7.2.4

Divide $- 3 x$ by $1$ .

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

Step 1.1.1.3

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

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Step 1.1.1.3.1

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

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

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 $n = 1$ .

$3 x^{2} - 3 x - 6 \cdot 1$

Step 1.1.1.3.3

Multiply $- 6$ by $1$ .

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

Set the first derivative equal to $0$ .

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

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $3$ out of $3 x^{2} - 3 x - 6$ .

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Step 1.2.2.1.1

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

$3 (x^{2}) - 3 x - 6 = 0$

Step 1.2.2.1.2

Factor $3$ out of $- 3 x$ .

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

Step 1.2.2.1.3

Factor $3$ out of $- 6$ .

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

Step 1.2.2.1.4

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

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

Step 1.2.2.1.5

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

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

Step 1.2.2.2

Factor.

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Step 1.2.2.2.1

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

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Step 1.2.2.2.1.1

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 $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 1.2.2.2.1.2

Write the factored form using these integers.

$3 ((x - 2) (x + 1)) = 0$

Step 1.2.2.2.2

Remove unnecessary parentheses.

$3 (x - 2) (x + 1) = 0$

Step 1.2.3

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

$x - 2 = 0$

$x + 1 = 0$

Step 1.2.4

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

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Step 1.2.4.1

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

$x - 2 = 0$

Step 1.2.4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.5

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 1.2.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.2.6

The final solution is all the values that make $3 (x - 2) (x + 1) = 0$ true.

$x = 2, - 1$

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 $x^{3} - \frac{3}{2} x^{2} - 6 x$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = 2$ .

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Step 1.4.1.1

Substitute $2$ for $x$ .

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

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

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

$8 - \frac{3}{2} \cdot {(2)}^{2} - 6 \cdot 2$

Step 1.4.1.2.1.2

Cancel the common factor of $2$ .

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Step 1.4.1.2.1.2.1

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

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

Step 1.4.1.2.1.2.2

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

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

Step 1.4.1.2.1.2.3

Cancel the common factor.

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

Step 1.4.1.2.1.2.4

Rewrite the expression.

$8 - 3 \cdot 2 - 6 \cdot 2$

Step 1.4.1.2.1.3

Multiply $- 3$ by $2$ .

$8 - 6 - 6 \cdot 2$

Step 1.4.1.2.1.4

Multiply $- 6$ by $2$ .

$8 - 6 - 12$

Step 1.4.1.2.2

Simplify by subtracting numbers.

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Step 1.4.1.2.2.1

Subtract $6$ from $8$ .

$2 - 12$

Step 1.4.1.2.2.2

Subtract $12$ from $2$ .

$- 10$

Step 1.4.2

Evaluate at $x = - 1$ .

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Step 1.4.2.1

Substitute $- 1$ for $x$ .

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

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

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

Step 1.4.2.2.1.2

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Step 1.4.2.2.1.2.1

Move ${(- 1)}^{2}$ .

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

Step 1.4.2.2.1.2.2

Multiply ${(- 1)}^{2}$ by $- 1$ .

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Step 1.4.2.2.1.2.2.1

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

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

Step 1.4.2.2.1.2.2.2

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

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

Step 1.4.2.2.1.2.3

Add $2$ and $1$ .

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

Step 1.4.2.2.1.3

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

$- 1 - \frac{3}{2} - 6 \cdot - 1$

Step 1.4.2.2.1.4

Multiply $- 6$ by $- 1$ .

$- 1 - \frac{3}{2} + 6$

Step 1.4.2.2.2

Find the common denominator.

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Step 1.4.2.2.2.1

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

$\frac{- 1}{1} - \frac{3}{2} + 6$

Step 1.4.2.2.2.2

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

$\frac{- 1}{1} \cdot \frac{2}{2} - \frac{3}{2} + 6$

Step 1.4.2.2.2.3

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

$\frac{- 1 \cdot 2}{2} - \frac{3}{2} + 6$

Step 1.4.2.2.2.4

Write $6$ as a fraction with denominator $1$ .

$\frac{- 1 \cdot 2}{2} - \frac{3}{2} + \frac{6}{1}$

Step 1.4.2.2.2.5

Multiply $\frac{6}{1}$ by $\frac{2}{2}$ .

$\frac{- 1 \cdot 2}{2} - \frac{3}{2} + \frac{6}{1} \cdot \frac{2}{2}$

Step 1.4.2.2.2.6

Multiply $\frac{6}{1}$ by $\frac{2}{2}$ .

$\frac{- 1 \cdot 2}{2} - \frac{3}{2} + \frac{6 \cdot 2}{2}$

Step 1.4.2.2.3

Combine the numerators over the common denominator.

$\frac{- 1 \cdot 2 - 3 + 6 \cdot 2}{2}$

Step 1.4.2.2.4

Simplify each term.

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Step 1.4.2.2.4.1

Multiply $- 1$ by $2$ .

$\frac{- 2 - 3 + 6 \cdot 2}{2}$

Step 1.4.2.2.4.2

Multiply $6$ by $2$ .

$\frac{- 2 - 3 + 12}{2}$

Step 1.4.2.2.5

Simplify by adding and subtracting.

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Step 1.4.2.2.5.1

Subtract $3$ from $- 2$ .

$\frac{- 5 + 12}{2}$

Step 1.4.2.2.5.2

Add $- 5$ and $12$ .

$\frac{7}{2}$

Step 1.4.3

List all of the points.

$(2, - 10), (- 1, \frac{7}{2})$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = - 2$ .

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Step 2.1.1

Substitute $- 2$ for $x$ .

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

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

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

$- 8 - \frac{3}{2} \cdot {(- 2)}^{2} - 6 \cdot - 2$

Step 2.1.2.1.2

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

$- 8 - \frac{3}{2} \cdot 4 - 6 \cdot - 2$

Step 2.1.2.1.3

Cancel the common factor of $2$ .

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Step 2.1.2.1.3.1

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

$- 8 + \frac{- 3}{2} \cdot 4 - 6 \cdot - 2$

Step 2.1.2.1.3.2

Factor $2$ out of $4$ .

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

Step 2.1.2.1.3.3

Cancel the common factor.

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

Step 2.1.2.1.3.4

Rewrite the expression.

$- 8 - 3 \cdot 2 - 6 \cdot - 2$

Step 2.1.2.1.4

Multiply $- 3$ by $2$ .

$- 8 - 6 - 6 \cdot - 2$

Step 2.1.2.1.5

Multiply $- 6$ by $- 2$ .

$- 8 - 6 + 12$

Step 2.1.2.2

Simplify by adding and subtracting.

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Step 2.1.2.2.1

Subtract $6$ from $- 8$ .

$- 14 + 12$

Step 2.1.2.2.2

Add $- 14$ and $12$ .

$- 2$

Step 2.2

Evaluate at $x = 3$ .

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Step 2.2.1

Substitute $3$ for $x$ .

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

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 $3$ to the power of $3$ .

$27 - \frac{3}{2} \cdot {(3)}^{2} - 6 \cdot 3$

Step 2.2.2.1.2

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

$27 - \frac{3}{2} \cdot 9 - 6 \cdot 3$

Step 2.2.2.1.3

Multiply $- \frac{3}{2} \cdot 9$ .

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Step 2.2.2.1.3.1

Multiply $9$ by $- 1$ .

$27 - 9 (\frac{3}{2}) - 6 \cdot 3$

Step 2.2.2.1.3.2

Combine $- 9$ and $\frac{3}{2}$ .

$27 + \frac{- 9 \cdot 3}{2} - 6 \cdot 3$

Step 2.2.2.1.3.3

Multiply $- 9$ by $3$ .

$27 + \frac{- 27}{2} - 6 \cdot 3$

Step 2.2.2.1.4

Move the negative in front of the fraction.

$27 - \frac{27}{2} - 6 \cdot 3$

Step 2.2.2.1.5

Multiply $- 6$ by $3$ .

$27 - \frac{27}{2} - 18$

Step 2.2.2.2

Find the common denominator.

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Step 2.2.2.2.1

Write $27$ as a fraction with denominator $1$ .

$\frac{27}{1} - \frac{27}{2} - 18$

Step 2.2.2.2.2

Multiply $\frac{27}{1}$ by $\frac{2}{2}$ .

$\frac{27}{1} \cdot \frac{2}{2} - \frac{27}{2} - 18$

Step 2.2.2.2.3

Multiply $\frac{27}{1}$ by $\frac{2}{2}$ .

$\frac{27 \cdot 2}{2} - \frac{27}{2} - 18$

Step 2.2.2.2.4

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

$\frac{27 \cdot 2}{2} - \frac{27}{2} + \frac{- 18}{1}$

Step 2.2.2.2.5

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

$\frac{27 \cdot 2}{2} - \frac{27}{2} + \frac{- 18}{1} \cdot \frac{2}{2}$

Step 2.2.2.2.6

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

$\frac{27 \cdot 2}{2} - \frac{27}{2} + \frac{- 18 \cdot 2}{2}$

Step 2.2.2.3

Combine the numerators over the common denominator.

$\frac{27 \cdot 2 - 27 - 18 \cdot 2}{2}$

Step 2.2.2.4

Simplify each term.

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Step 2.2.2.4.1

Multiply $27$ by $2$ .

$\frac{54 - 27 - 18 \cdot 2}{2}$

Step 2.2.2.4.2

Multiply $- 18$ by $2$ .

$\frac{54 - 27 - 36}{2}$

Step 2.2.2.5

Simplify the expression.

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Step 2.2.2.5.1

Subtract $27$ from $54$ .

$\frac{27 - 36}{2}$

Step 2.2.2.5.2

Subtract $36$ from $27$ .

$\frac{- 9}{2}$

Step 2.2.2.5.3

Move the negative in front of the fraction.

$- \frac{9}{2}$

Step 2.3

List all of the points.

$(- 2, - 2), (3, - \frac{9}{2})$

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: $(- 1, \frac{7}{2})$

Absolute Minimum: $(2, - 10)$

Step 4