Calculus Examples

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

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

Differentiate.

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Step 1.1.1.1

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

$\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [- 24 x]$

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 $n = 3$ .

$3 x^{2} + \frac{d}{d x} [- 6 x^{2}] + \frac{d}{d x} [- 24 x]$

Step 1.1.2

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

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Step 1.1.2.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x^{2}$ with respect to $x$ is $- 6 \frac{d}{d x} [x^{2}]$ .

$3 x^{2} - 6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 24 x]$

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 $n = 2$ .

$3 x^{2} - 6 (2 x) + \frac{d}{d x} [- 24 x]$

Step 1.1.2.3

Multiply $2$ by $- 6$ .

$3 x^{2} - 12 x + \frac{d}{d x} [- 24 x]$

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

$3 x^{2} - 12 x - 24 \frac{d}{d x} [x]$

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 $n = 1$ .

$3 x^{2} - 12 x - 24 \cdot 1$

Step 1.1.3.3

Multiply $- 24$ by $1$ .

$f' (x) = 3 x^{2} - 12 x - 24$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2} - 12 x - 24$ .

$3 x^{2} - 12 x - 24$

Step 2

Set the first derivative equal to $0$ then solve the equation $3 x^{2} - 12 x - 24 = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$3 x^{2} - 12 x - 24 = 0$

Step 2.2

Factor $3$ out of $3 x^{2} - 12 x - 24$ .

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Step 2.2.1

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

$3 (x^{2}) - 12 x - 24 = 0$

Step 2.2.2

Factor $3$ out of $- 12 x$ .

$3 (x^{2}) + 3 (- 4 x) - 24 = 0$

Step 2.2.3

Factor $3$ out of $- 24$ .

$3 x^{2} + 3 (- 4 x) + 3 \cdot - 8 = 0$

Step 2.2.4

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

$3 (x^{2} - 4 x) + 3 \cdot - 8 = 0$

Step 2.2.5

Factor $3$ out of $3 (x^{2} - 4 x) + 3 \cdot - 8$ .

$3 (x^{2} - 4 x - 8) = 0$

Step 2.3

Divide each term in $3 (x^{2} - 4 x - 8) = 0$ by $3$ and simplify.

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Step 2.3.1

Divide each term in $3 (x^{2} - 4 x - 8) = 0$ by $3$ .

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $3$ .

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Step 2.3.2.1.1

Cancel the common factor.

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

Step 2.3.2.1.2

Divide $x^{2} - 4 x - 8$ by $1$ .

$x^{2} - 4 x - 8 = \frac{0}{3}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $0$ by $3$ .

$x^{2} - 4 x - 8 = 0$

Step 2.4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.5

Substitute the values $a = 1$ , $b = - 4$ , and $c = - 8$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot - 8)}}{2 \cdot 1}$

Step 2.6

Simplify.

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 8}}{2 \cdot 1}$

Step 2.6.1.2

Multiply $- 4 \cdot 1 \cdot - 8$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 8}}{2 \cdot 1}$

Step 2.6.1.2.2

Multiply $- 4$ by $- 8$ .

$x = \frac{4 \pm \sqrt{16 + 32}}{2 \cdot 1}$

Step 2.6.1.3

Add $16$ and $32$ .

$x = \frac{4 \pm \sqrt{48}}{2 \cdot 1}$

Step 2.6.1.4

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 2.6.1.4.1

Factor $16$ out of $48$ .

$x = \frac{4 \pm \sqrt{16 (3)}}{2 \cdot 1}$

Step 2.6.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{4 \pm \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 2.6.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 4 \sqrt{3}}{2 \cdot 1}$

Step 2.6.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 4 \sqrt{3}}{2}$

Step 2.6.3

Simplify $\frac{4 \pm 4 \sqrt{3}}{2}$ .

$x = 2 \pm 2 \sqrt{3}$

Step 2.7

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 8}}{2 \cdot 1}$

Step 2.7.1.2

Multiply $- 4 \cdot 1 \cdot - 8$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 8}}{2 \cdot 1}$

Step 2.7.1.2.2

Multiply $- 4$ by $- 8$ .

$x = \frac{4 \pm \sqrt{16 + 32}}{2 \cdot 1}$

Step 2.7.1.3

Add $16$ and $32$ .

$x = \frac{4 \pm \sqrt{48}}{2 \cdot 1}$

Step 2.7.1.4

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 2.7.1.4.1

Factor $16$ out of $48$ .

$x = \frac{4 \pm \sqrt{16 (3)}}{2 \cdot 1}$

Step 2.7.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{4 \pm \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 2.7.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 4 \sqrt{3}}{2 \cdot 1}$

Step 2.7.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 4 \sqrt{3}}{2}$

Step 2.7.3

Simplify $\frac{4 \pm 4 \sqrt{3}}{2}$ .

$x = 2 \pm 2 \sqrt{3}$

Step 2.7.4

Change the $\pm$ to $+$ .

$x = 2 + 2 \sqrt{3}$

Step 2.8

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 2.8.1

Simplify the numerator.

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Step 2.8.1.1

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 8}}{2 \cdot 1}$

Step 2.8.1.2

Multiply $- 4 \cdot 1 \cdot - 8$ .

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Step 2.8.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 8}}{2 \cdot 1}$

Step 2.8.1.2.2

Multiply $- 4$ by $- 8$ .

$x = \frac{4 \pm \sqrt{16 + 32}}{2 \cdot 1}$

Step 2.8.1.3

Add $16$ and $32$ .

$x = \frac{4 \pm \sqrt{48}}{2 \cdot 1}$

Step 2.8.1.4

Rewrite $48$ as $4^{2} \cdot 3$ .

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Step 2.8.1.4.1

Factor $16$ out of $48$ .

$x = \frac{4 \pm \sqrt{16 (3)}}{2 \cdot 1}$

Step 2.8.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{4 \pm \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 2.8.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 4 \sqrt{3}}{2 \cdot 1}$

Step 2.8.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 4 \sqrt{3}}{2}$

Step 2.8.3

Simplify $\frac{4 \pm 4 \sqrt{3}}{2}$ .

$x = 2 \pm 2 \sqrt{3}$

Step 2.8.4

Change the $\pm$ to $-$ .

$x = 2 - 2 \sqrt{3}$

Step 2.9

The final answer is the combination of both solutions.

$x = 2 + 2 \sqrt{3}, 2 - 2 \sqrt{3}$

Step 3

The values which make the derivative equal to $0$ are $2 + 2 \sqrt{3}, 2 - 2 \sqrt{3}$ .

$2 + 2 \sqrt{3}, 2 - 2 \sqrt{3}$

Step 4

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

$(- \infty, 2 - 2 \sqrt{3}) \cup (2 - 2 \sqrt{3}, 2 + 2 \sqrt{3}) \cup (2 + 2 \sqrt{3}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, 2 - 2 \sqrt{3})$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 2.4641017$ in the expression.

$f' (- 2.4641017) = 3 {(- 2.4641017)}^{2} - 12 \cdot - 2.4641017 - 24$

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 $- 2.4641017$ to the power of $2$ .

$f' (- 2.4641017) = 3 \cdot 6.07179718 - 12 \cdot - 2.4641017 - 24$

Step 5.2.1.2

Multiply $3$ by $6.07179718$ .

$f' (- 2.4641017) = 18.21539156 - 12 \cdot - 2.4641017 - 24$

Step 5.2.1.3

Multiply $- 12$ by $- 2.4641017$ .

$f' (- 2.4641017) = 18.21539156 + 29.5692204 - 24$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $18.21539156$ and $29.5692204$ .

$f' (- 2.4641017) = 47.78461196 - 24$

Step 5.2.2.2

Subtract $24$ from $47.78461196$ .

$f' (- 2.4641017) = 23.78461196$

Step 5.2.3

The final answer is $23.78461196$ .

$23.78461196$

Step 5.3

At $x = - 2.4641017$ the derivative is $23.78461196$ . Since this is positive, the function is increasing on $(- \infty, 2 - 2 \sqrt{3})$ .

Increasing on $(- \infty, 2 - 2 \sqrt{3})$ since $f' (x) > 0$

Step 6

Substitute a value from the interval $(- 1.4641017, 2 + 2 \sqrt{3})$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $2.00000015$ in the expression.

$f' (2.00000015) = 3 {(2.00000015)}^{2} - 12 \cdot 2.00000015 - 24$

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 $2.00000015$ to the power of $2$ .

$f' (2.00000015) = 3 \cdot 4.0000006 - 12 \cdot 2.00000015 - 24$

Step 6.2.1.2

Multiply $3$ by $4.0000006$ .

$f' (2.00000015) = 12.0000018 - 12 \cdot 2.00000015 - 24$

Step 6.2.1.3

Multiply $- 12$ by $2.00000015$ .

$f' (2.00000015) = 12.0000018 - 24.0000018 - 24$

Step 6.2.2

Simplify by subtracting numbers.

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Step 6.2.2.1

Subtract $24.0000018$ from $12.0000018$ .

$f' (2.00000015) = - 12 - 24$

Step 6.2.2.2

Subtract $24$ from $- 12$ .

$f' (2.00000015) = - 36$

Step 6.2.3

The final answer is $- 36$ .

$- 36$

Step 6.3

At $x = 2.00000015$ the derivative is $- 36$ . Since this is negative, the function is decreasing on $(- 1.4641017, 2 + 2 \sqrt{3})$ .

Decreasing on $(2 - 2 \sqrt{3}, 2 + 2 \sqrt{3})$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(2 + 2 \sqrt{3}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $6.464102$ in the expression.

$f' (6.464102) = 3 {(6.464102)}^{2} - 12 \cdot 6.464102 - 24$

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

Raise $6.464102$ to the power of $2$ .

$f' (6.464102) = 3 \cdot 41.78461466 - 12 \cdot 6.464102 - 24$

Step 7.2.1.2

Multiply $3$ by $41.78461466$ .

$f' (6.464102) = 125.35384399 - 12 \cdot 6.464102 - 24$

Step 7.2.1.3

Multiply $- 12$ by $6.464102$ .

$f' (6.464102) = 125.35384399 - 77.569224 - 24$

Step 7.2.2

Simplify by subtracting numbers.

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Step 7.2.2.1

Subtract $77.569224$ from $125.35384399$ .

$f' (6.464102) = 47.78461999 - 24$

Step 7.2.2.2

Subtract $24$ from $47.78461999$ .

$f' (6.464102) = 23.78461999$

Step 7.2.3

The final answer is $23.78461999$ .

$23.78461999$

Step 7.3

At $x = 6.464102$ the derivative is $23.78461999$ . Since this is positive, the function is increasing on $(2 + 2 \sqrt{3}, \infty)$ .

Increasing on $(2 + 2 \sqrt{3}, \infty)$ since $f' (x) > 0$

Step 8

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

Increasing on: $(- \infty, 2 - 2 \sqrt{3}), (2 + 2 \sqrt{3}, \infty)$

Decreasing on: $(2 - 2 \sqrt{3}, 2 + 2 \sqrt{3})$

Step 9