Calculus Examples

$f (x) = 2 x^{3} - 21 x^{2} - 48 x$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

By the Sum Rule, the derivative of $2 x^{3} - 21 x^{2} - 48 x$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- 21 x^{2}] + \frac{d}{d x} [- 48 x]$ .

$f' (x) = \frac{d}{d x} (2 x^{3}) + \frac{d}{d x} (- 21 x^{2}) + \frac{d}{d x} (- 48 x)$

Step 1.1.2

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

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Step 1.1.2.1

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

$f' (x) = 2 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 21 x^{2}) + \frac{d}{d x} (- 48 x)$

Step 1.1.2.2

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

$f' (x) = 2 (3 x^{2}) + \frac{d}{d x} (- 21 x^{2}) + \frac{d}{d x} (- 48 x)$

Step 1.1.2.3

Multiply $3$ by $2$ .

$f' (x) = 6 x^{2} + \frac{d}{d x} (- 21 x^{2}) + \frac{d}{d x} (- 48 x)$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 21 x^{2}]$ .

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Step 1.1.3.1

Since $- 21$ is constant with respect to $x$ , the derivative of $- 21 x^{2}$ with respect to $x$ is $- 21 \frac{d}{d x} [x^{2}]$ .

$f' (x) = 6 x^{2} - 21 \frac{d}{d x} x^{2} + \frac{d}{d x} (- 48 x)$

Step 1.1.3.2

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

$f' (x) = 6 x^{2} - 21 (2 x) + \frac{d}{d x} (- 48 x)$

Step 1.1.3.3

Multiply $2$ by $- 21$ .

$f' (x) = 6 x^{2} - 42 x + \frac{d}{d x} (- 48 x)$

Step 1.1.4

Evaluate $\frac{d}{d x} [- 48 x]$ .

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Step 1.1.4.1

Since $- 48$ is constant with respect to $x$ , the derivative of $- 48 x$ with respect to $x$ is $- 48 \frac{d}{d x} [x]$ .

$f' (x) = 6 x^{2} - 42 x - 48 \frac{d}{d x} x$

Step 1.1.4.2

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

$f' (x) = 6 x^{2} - 42 x - 48 \cdot 1$

Step 1.1.4.3

Multiply $- 48$ by $1$ .

$f' (x) = 6 x^{2} - 42 x - 48$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $6 x^{2} - 42 x - 48$ .

$6 x^{2} - 42 x - 48$

Step 2

Set the first derivative equal to $0$ then solve the equation $6 x^{2} - 42 x - 48 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$6 x^{2} - 42 x - 48 = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $6$ out of $6 x^{2} - 42 x - 48$ .

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Step 2.2.1.1

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

$6 (x^{2}) - 42 x - 48 = 0$

Step 2.2.1.2

Factor $6$ out of $- 42 x$ .

$6 (x^{2}) + 6 (- 7 x) - 48 = 0$

Step 2.2.1.3

Factor $6$ out of $- 48$ .

$6 x^{2} + 6 (- 7 x) + 6 \cdot - 8 = 0$

Step 2.2.1.4

Factor $6$ out of $6 x^{2} + 6 (- 7 x)$ .

$6 (x^{2} - 7 x) + 6 \cdot - 8 = 0$

Step 2.2.1.5

Factor $6$ out of $6 (x^{2} - 7 x) + 6 \cdot - 8$ .

$6 (x^{2} - 7 x - 8) = 0$

Step 2.2.2

Factor.

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Step 2.2.2.1

Factor $x^{2} - 7 x - 8$ using the AC method.

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Step 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 $- 8$ and whose sum is $- 7$ .

$- 8, 1$

Step 2.2.2.1.2

Write the factored form using these integers.

$6 ((x - 8) (x + 1)) = 0$

Step 2.2.2.2

Remove unnecessary parentheses.

$6 (x - 8) (x + 1) = 0$

Step 2.3

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

$x - 8 = 0$

$x + 1 = 0$

Step 2.4

Set $x - 8$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 2.4.2

Add $8$ to both sides of the equation.

$x = 8$

Step 2.5

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

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Step 2.5.1

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

$x + 1 = 0$

Step 2.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.6

The final solution is all the values that make $6 (x - 8) (x + 1) = 0$ true.

$x = 8, - 1$

Step 3

The values which make the derivative equal to $0$ are $8, - 1$ .

$8, - 1$

Step 4

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

$(- \infty, - 1) \cup (- 1, 8) \cup (8, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 1)$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

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

$f' (- 2) = 6 {(- 2)}^{2} - 42 \cdot - 2 - 48$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

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

$f' (- 2) = 6 \cdot 4 - 42 \cdot - 2 - 48$

Step 5.2.1.2

Multiply $6$ by $4$ .

$f' (- 2) = 24 - 42 \cdot - 2 - 48$

Step 5.2.1.3

Multiply $- 42$ by $- 2$ .

$f' (- 2) = 24 + 84 - 48$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $24$ and $84$ .

$f' (- 2) = 108 - 48$

Step 5.2.2.2

Subtract $48$ from $108$ .

$f' (- 2) = 60$

Step 5.2.3

The final answer is $60$ .

$60$

Step 5.3

At $x = - 2$ the derivative is $60$ . Since this is positive, the function is increasing on $(- \infty, - 1)$ .

Increasing on $(- \infty, - 1)$ since $f' (x) > 0$

Step 6

Substitute a value from the interval $(- 1, 8)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $\frac{7}{2}$ in the expression.

$f' (\frac{7}{2}) = 6 {(\frac{7}{2})}^{2} - 42 (\frac{7}{2}) - 48$

Step 6.2

Simplify the result.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Apply the product rule to $\frac{7}{2}$ .

$f' (\frac{7}{2}) = 6 (\frac{7^{2}}{2^{2}}) - 42 (\frac{7}{2}) - 48$

Step 6.2.1.2

Raise $7$ to the power of $2$ .

$f' (\frac{7}{2}) = 6 (\frac{49}{2^{2}}) - 42 (\frac{7}{2}) - 48$

Step 6.2.1.3

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

$f' (\frac{7}{2}) = 6 (\frac{49}{4}) - 42 (\frac{7}{2}) - 48$

Step 6.2.1.4

Cancel the common factor of $2$ .

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Step 6.2.1.4.1

Factor $2$ out of $6$ .

$f' (\frac{7}{2}) = 2 (3) (\frac{49}{4}) - 42 (\frac{7}{2}) - 48$

Step 6.2.1.4.2

Factor $2$ out of $4$ .

$f' (\frac{7}{2}) = 2 \cdot (3 (\frac{49}{2 \cdot 2})) - 42 (\frac{7}{2}) - 48$

Step 6.2.1.4.3

Cancel the common factor.

$f' (\frac{7}{2}) = 2 \cdot (3 (\frac{49}{2 \cdot 2})) - 42 (\frac{7}{2}) - 48$

Step 6.2.1.4.4

Rewrite the expression.

$f' (\frac{7}{2}) = 3 (\frac{49}{2}) - 42 (\frac{7}{2}) - 48$

Step 6.2.1.5

Combine $3$ and $\frac{49}{2}$ .

$f' (\frac{7}{2}) = \frac{3 \cdot 49}{2} - 42 (\frac{7}{2}) - 48$

Step 6.2.1.6

Multiply $3$ by $49$ .

$f' (\frac{7}{2}) = \frac{147}{2} - 42 (\frac{7}{2}) - 48$

Step 6.2.1.7

Cancel the common factor of $2$ .

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Step 6.2.1.7.1

Factor $2$ out of $- 42$ .

$f' (\frac{7}{2}) = \frac{147}{2} + 2 (- 21) (\frac{7}{2}) - 48$

Step 6.2.1.7.2

Cancel the common factor.

$f' (\frac{7}{2}) = \frac{147}{2} + 2 \cdot (- 21 (\frac{7}{2})) - 48$

Step 6.2.1.7.3

Rewrite the expression.

$f' (\frac{7}{2}) = \frac{147}{2} - 21 \cdot 7 - 48$

Step 6.2.1.8

Multiply $- 21$ by $7$ .

$f' (\frac{7}{2}) = \frac{147}{2} - 147 - 48$

Step 6.2.2

Find the common denominator.

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Step 6.2.2.1

Write $- 147$ as a fraction with denominator $1$ .

$f' (\frac{7}{2}) = \frac{147}{2} + \frac{- 147}{1} - 48$

Step 6.2.2.2

Multiply $\frac{- 147}{1}$ by $\frac{2}{2}$ .

$f' (\frac{7}{2}) = \frac{147}{2} + \frac{- 147}{1} \cdot \frac{2}{2} - 48$

Step 6.2.2.3

Multiply $\frac{- 147}{1}$ by $\frac{2}{2}$ .

$f' (\frac{7}{2}) = \frac{147}{2} + \frac{- 147 \cdot 2}{2} - 48$

Step 6.2.2.4

Write $- 48$ as a fraction with denominator $1$ .

$f' (\frac{7}{2}) = \frac{147}{2} + \frac{- 147 \cdot 2}{2} + \frac{- 48}{1}$

Step 6.2.2.5

Multiply $\frac{- 48}{1}$ by $\frac{2}{2}$ .

$f' (\frac{7}{2}) = \frac{147}{2} + \frac{- 147 \cdot 2}{2} + \frac{- 48}{1} \cdot \frac{2}{2}$

Step 6.2.2.6

Multiply $\frac{- 48}{1}$ by $\frac{2}{2}$ .

$f' (\frac{7}{2}) = \frac{147}{2} + \frac{- 147 \cdot 2}{2} + \frac{- 48 \cdot 2}{2}$

Step 6.2.3

Combine the numerators over the common denominator.

$f' (\frac{7}{2}) = \frac{147 - 147 \cdot 2 - 48 \cdot 2}{2}$

Step 6.2.4

Simplify each term.

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Step 6.2.4.1

Multiply $- 147$ by $2$ .

$f' (\frac{7}{2}) = \frac{147 - 294 - 48 \cdot 2}{2}$

Step 6.2.4.2

Multiply $- 48$ by $2$ .

$f' (\frac{7}{2}) = \frac{147 - 294 - 96}{2}$

Step 6.2.5

Simplify the expression.

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Step 6.2.5.1

Subtract $294$ from $147$ .

$f' (\frac{7}{2}) = \frac{- 147 - 96}{2}$

Step 6.2.5.2

Subtract $96$ from $- 147$ .

$f' (\frac{7}{2}) = \frac{- 243}{2}$

Step 6.2.5.3

Move the negative in front of the fraction.

$f' (\frac{7}{2}) = - \frac{243}{2}$

Step 6.2.6

The final answer is $- \frac{243}{2}$ .

$- \frac{243}{2}$

Step 6.3

At $x = \frac{7}{2}$ the derivative is $- \frac{243}{2}$ . Since this is negative, the function is decreasing on $(- 1, 8)$ .

Decreasing on $(- 1, 8)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(8, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $9$ in the expression.

$f' (9) = 6 {(9)}^{2} - 42 \cdot 9 - 48$

Step 7.2

Simplify the result.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Raise $9$ to the power of $2$ .

$f' (9) = 6 \cdot 81 - 42 \cdot 9 - 48$

Step 7.2.1.2

Multiply $6$ by $81$ .

$f' (9) = 486 - 42 \cdot 9 - 48$

Step 7.2.1.3

Multiply $- 42$ by $9$ .

$f' (9) = 486 - 378 - 48$

Step 7.2.2

Simplify by subtracting numbers.

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Step 7.2.2.1

Subtract $378$ from $486$ .

$f' (9) = 108 - 48$

Step 7.2.2.2

Subtract $48$ from $108$ .

$f' (9) = 60$

Step 7.2.3

The final answer is $60$ .

$60$

Step 7.3

At $x = 9$ the derivative is $60$ . Since this is positive, the function is increasing on $(8, \infty)$ .

Increasing on $(8, \infty)$ since $f' (x) > 0$

Step 8

List the intervals on which the function is increasing and decreasing.

Increasing on: $(- \infty, - 1), (8, \infty)$

Decreasing on: $(- 1, 8)$

Step 9