Calculus Examples

$f (x) = x^{3} + 3 x^{2} - 24 x + 6$

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

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of $x^{3} + 3 x^{2} - 24 x + 6$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [6]$ .

$f' (x) = \frac{d}{d x} (x^{3}) + \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- 24 x) + \frac{d}{d x} (6)$

Step 1.1.1.2

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

$f' (x) = 3 x^{2} + \frac{d}{d x} (3 x^{2}) + \frac{d}{d x} (- 24 x) + \frac{d}{d x} (6)$

Step 1.1.2

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

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Step 1.1.2.1

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

$f' (x) = 3 x^{2} + 3 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 24 x) + \frac{d}{d x} (6)$

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 $n = 2$ .

$f' (x) = 3 x^{2} + 3 (2 x) + \frac{d}{d x} (- 24 x) + \frac{d}{d x} (6)$

Step 1.1.2.3

Multiply $2$ by $3$ .

$f' (x) = 3 x^{2} + 6 x + \frac{d}{d x} (- 24 x) + \frac{d}{d x} (6)$

Step 1.1.3

Evaluate $\frac{d}{d x} [- 24 x]$ .

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Step 1.1.3.1

Since $- 24$ is constant with respect to $x$ , the derivative of $- 24 x$ with respect to $x$ is $- 24 \frac{d}{d x} [x]$ .

$f' (x) = 3 x^{2} + 6 x - 24 \frac{d}{d x} x + \frac{d}{d x} (6)$

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 $n = 1$ .

$f' (x) = 3 x^{2} + 6 x - 24 \cdot 1 + \frac{d}{d x} (6)$

Step 1.1.3.3

Multiply $- 24$ by $1$ .

$f' (x) = 3 x^{2} + 6 x - 24 + \frac{d}{d x} (6)$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

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

$f' (x) = 3 x^{2} + 6 x - 24 + 0$

Step 1.1.4.2

Add $3 x^{2} + 6 x - 24$ and $0$ .

$f' (x) = 3 x^{2} + 6 x - 24$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2} + 6 x - 24$ .

$3 x^{2} + 6 x - 24$

Step 2

Set the first derivative equal to $0$ then solve the equation $3 x^{2} + 6 x - 24 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$3 x^{2} + 6 x - 24 = 0$

Step 2.2

Factor the left side of the equation.

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Step 2.2.1

Factor $3$ out of $3 x^{2} + 6 x - 24$ .

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Step 2.2.1.1

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

$3 (x^{2}) + 6 x - 24 = 0$

Step 2.2.1.2

Factor $3$ out of $6 x$ .

$3 (x^{2}) + 3 (2 x) - 24 = 0$

Step 2.2.1.3

Factor $3$ out of $- 24$ .

$3 x^{2} + 3 (2 x) + 3 \cdot - 8 = 0$

Step 2.2.1.4

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

$3 (x^{2} + 2 x) + 3 \cdot - 8 = 0$

Step 2.2.1.5

Factor $3$ out of $3 (x^{2} + 2 x) + 3 \cdot - 8$ .

$3 (x^{2} + 2 x - 8) = 0$

Step 2.2.2

Factor.

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Step 2.2.2.1

Factor $x^{2} + 2 x - 8$ 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 $2$ .

$- 2, 4$

Step 2.2.2.1.2

Write the factored form using these integers.

$3 ((x - 2) (x + 4)) = 0$

Step 2.2.2.2

Remove unnecessary parentheses.

$3 (x - 2) (x + 4) = 0$

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 $0$ .

$x - 2 = 0$

$x + 4 = 0$

Step 2.4

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

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Step 2.4.1

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

$x - 2 = 0$

Step 2.4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.5

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

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Step 2.5.1

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

$x + 4 = 0$

Step 2.5.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 2.6

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

$x = 2, - 4$

Step 3

The values which make the derivative equal to $0$ are $2, - 4$ .

$2, - 4$

Step 4

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

$(- \infty, - 4) \cup (- 4, 2) \cup (2, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 4)$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 5$ in the expression.

$f' (- 5) = 3 {(- 5)}^{2} + 6 (- 5) - 24$

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

$f' (- 5) = 3 \cdot 25 + 6 (- 5) - 24$

Step 5.2.1.2

Multiply $3$ by $25$ .

$f' (- 5) = 75 + 6 (- 5) - 24$

Step 5.2.1.3

Multiply $6$ by $- 5$ .

$f' (- 5) = 75 - 30 - 24$

Step 5.2.2

Simplify by subtracting numbers.

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Step 5.2.2.1

Subtract $30$ from $75$ .

$f' (- 5) = 45 - 24$

Step 5.2.2.2

Subtract $24$ from $45$ .

$f' (- 5) = 21$

Step 5.2.3

The final answer is $21$ .

$21$

Step 5.3

At $x = - 5$ the derivative is $21$ . Since this is positive, the function is increasing on $(- \infty, - 4)$ .

Increasing on $(- \infty, - 4)$ since $f' (x) > 0$

Step 6

Substitute a value from the interval $(- 4, 2)$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $- 1$ in the expression.

$f' (- 1) = 3 {(- 1)}^{2} + 6 (- 1) - 24$

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

Raise $- 1$ to the power of $2$ .

$f' (- 1) = 3 \cdot 1 + 6 (- 1) - 24$

Step 6.2.1.2

Multiply $3$ by $1$ .

$f' (- 1) = 3 + 6 (- 1) - 24$

Step 6.2.1.3

Multiply $6$ by $- 1$ .

$f' (- 1) = 3 - 6 - 24$

Step 6.2.2

Simplify by subtracting numbers.

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Step 6.2.2.1

Subtract $6$ from $3$ .

$f' (- 1) = - 3 - 24$

Step 6.2.2.2

Subtract $24$ from $- 3$ .

$f' (- 1) = - 27$

Step 6.2.3

The final answer is $- 27$ .

$- 27$

Step 6.3

At $x = - 1$ the derivative is $- 27$ . Since this is negative, the function is decreasing on $(- 4, 2)$ .

Decreasing on $(- 4, 2)$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(2, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

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

$f' (3) = 3 {(3)}^{2} + 6 (3) - 24$

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

Multiply $3$ by ${(3)}^{2}$ by adding the exponents.

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Step 7.2.1.1.1

Multiply $3$ by ${(3)}^{2}$ .

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Step 7.2.1.1.1.1

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

$f' (3) = 3 {(3)}^{2} + 6 (3) - 24$

Step 7.2.1.1.1.2

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

$f' (3) = 3^{1 + 2} + 6 (3) - 24$

Step 7.2.1.1.2

Add $1$ and $2$ .

$f' (3) = 3^{3} + 6 (3) - 24$

Step 7.2.1.2

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

$f' (3) = 27 + 6 (3) - 24$

Step 7.2.1.3

Multiply $6$ by $3$ .

$f' (3) = 27 + 18 - 24$

Step 7.2.2

Simplify by adding and subtracting.

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Step 7.2.2.1

Add $27$ and $18$ .

$f' (3) = 45 - 24$

Step 7.2.2.2

Subtract $24$ from $45$ .

$f' (3) = 21$

Step 7.2.3

The final answer is $21$ .

$21$

Step 7.3

At $x = 3$ the derivative is $21$ . Since this is positive, the function is increasing on $(2, \infty)$ .

Increasing on $(2, \infty)$ since $f' (x) > 0$

Step 8

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

Increasing on: $(- \infty, - 4), (2, \infty)$

Decreasing on: $(- 4, 2)$

Step 9