Calculus Examples

$f (x) = - 2 x - \frac{1}{2} x^{2} + \frac{1}{2} x^{3}$

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 $- 2 x - \frac{1}{2} x^{2} + \frac{1}{2} x^{3}$ with respect to $x$ is $\frac{d}{d x} [- 2 x] + \frac{d}{d x} [- \frac{1}{2} x^{2}] + \frac{d}{d x} [\frac{1}{2} x^{3}]$ .

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

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 $n = 1$ .

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

Step 1.1.2.3

Multiply $- 2$ by $1$ .

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

Step 1.1.3

Evaluate $\frac{d}{d x} [- \frac{1}{2} x^{2}]$ .

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Step 1.1.3.1

Since $- \frac{1}{2}$ is constant with respect to $x$ , the derivative of $- \frac{1}{2} x^{2}$ with respect to $x$ is $- \frac{1}{2} \frac{d}{d x} [x^{2}]$ .

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

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 $n = 2$ .

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

Step 1.1.3.3

Multiply $2$ by $- 1$ .

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

Step 1.1.3.4

Combine $- 2$ and $\frac{1}{2}$ .

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

Step 1.1.3.5

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

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

Step 1.1.3.6

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

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Step 1.1.3.6.1

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

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

Step 1.1.3.6.2

Cancel the common factors.

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Step 1.1.3.6.2.1

Factor $2$ out of $2$ .

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

Step 1.1.3.6.2.2

Cancel the common factor.

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

Step 1.1.3.6.2.3

Rewrite the expression.

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

Step 1.1.3.6.2.4

Divide $- x$ by $1$ .

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

Step 1.1.4

Evaluate $\frac{d}{d x} [\frac{1}{2} x^{3}]$ .

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Step 1.1.4.1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{1}{2} x^{3}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x^{3}]$ .

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

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 $n = 3$ .

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

Step 1.1.4.3

Combine $3$ and $\frac{1}{2}$ .

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

Step 1.1.4.4

Combine $\frac{3}{2}$ and $x^{2}$ .

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

Step 1.1.5

Reorder terms.

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

Step 1.2

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

$\frac{3 x^{2}}{2} - x - 2$

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Multiply through by the least common denominator $2$ , then simplify.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Simplify.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

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

Step 2.2.2.1.2

Rewrite the expression.

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

Step 2.2.2.2

Multiply $- 1$ by $2$ .

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

Step 2.2.2.3

Multiply $2$ by $- 2$ .

$3 x^{2} - 2 x - 4 = 0$

Step 2.3

Use the quadratic formula to find the solutions.

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

Step 2.4

Substitute the values $a = 3$ , $b = - 2$ , and $c = - 4$ into the quadratic formula and solve for $x$ .

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

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

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

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

Step 2.5.1.2

Multiply $- 4 \cdot 3 \cdot - 4$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12 \cdot - 4}}{2 \cdot 3}$

Step 2.5.1.2.2

Multiply $- 12$ by $- 4$ .

$x = \frac{2 \pm \sqrt{4 + 48}}{2 \cdot 3}$

Step 2.5.1.3

Add $4$ and $48$ .

$x = \frac{2 \pm \sqrt{52}}{2 \cdot 3}$

Step 2.5.1.4

Rewrite $52$ as $2^{2} \cdot 13$ .

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Step 2.5.1.4.1

Factor $4$ out of $52$ .

$x = \frac{2 \pm \sqrt{4 (13)}}{2 \cdot 3}$

Step 2.5.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 13}}{2 \cdot 3}$

Step 2.5.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{13}}{2 \cdot 3}$

Step 2.5.2

Multiply $2$ by $3$ .

$x = \frac{2 \pm 2 \sqrt{13}}{6}$

Step 2.5.3

Simplify $\frac{2 \pm 2 \sqrt{13}}{6}$ .

$x = \frac{1 \pm \sqrt{13}}{3}$

Step 2.6

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

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

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

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

Step 2.6.1.2

Multiply $- 4 \cdot 3 \cdot - 4$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12 \cdot - 4}}{2 \cdot 3}$

Step 2.6.1.2.2

Multiply $- 12$ by $- 4$ .

$x = \frac{2 \pm \sqrt{4 + 48}}{2 \cdot 3}$

Step 2.6.1.3

Add $4$ and $48$ .

$x = \frac{2 \pm \sqrt{52}}{2 \cdot 3}$

Step 2.6.1.4

Rewrite $52$ as $2^{2} \cdot 13$ .

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Step 2.6.1.4.1

Factor $4$ out of $52$ .

$x = \frac{2 \pm \sqrt{4 (13)}}{2 \cdot 3}$

Step 2.6.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 13}}{2 \cdot 3}$

Step 2.6.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{13}}{2 \cdot 3}$

Step 2.6.2

Multiply $2$ by $3$ .

$x = \frac{2 \pm 2 \sqrt{13}}{6}$

Step 2.6.3

Simplify $\frac{2 \pm 2 \sqrt{13}}{6}$ .

$x = \frac{1 \pm \sqrt{13}}{3}$

Step 2.6.4

Change the $\pm$ to $+$ .

$x = \frac{1 + \sqrt{13}}{3}$

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

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

Step 2.7.1.2

Multiply $- 4 \cdot 3 \cdot - 4$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $3$ .

$x = \frac{2 \pm \sqrt{4 - 12 \cdot - 4}}{2 \cdot 3}$

Step 2.7.1.2.2

Multiply $- 12$ by $- 4$ .

$x = \frac{2 \pm \sqrt{4 + 48}}{2 \cdot 3}$

Step 2.7.1.3

Add $4$ and $48$ .

$x = \frac{2 \pm \sqrt{52}}{2 \cdot 3}$

Step 2.7.1.4

Rewrite $52$ as $2^{2} \cdot 13$ .

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Step 2.7.1.4.1

Factor $4$ out of $52$ .

$x = \frac{2 \pm \sqrt{4 (13)}}{2 \cdot 3}$

Step 2.7.1.4.2

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 13}}{2 \cdot 3}$

Step 2.7.1.5

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{13}}{2 \cdot 3}$

Step 2.7.2

Multiply $2$ by $3$ .

$x = \frac{2 \pm 2 \sqrt{13}}{6}$

Step 2.7.3

Simplify $\frac{2 \pm 2 \sqrt{13}}{6}$ .

$x = \frac{1 \pm \sqrt{13}}{3}$

Step 2.7.4

Change the $\pm$ to $-$ .

$x = \frac{1 - \sqrt{13}}{3}$

Step 2.8

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{13}}{3}, \frac{1 - \sqrt{13}}{3}$

Step 3

The values which make the derivative equal to $0$ are $\frac{1 + \sqrt{13}}{3}, \frac{1 - \sqrt{13}}{3}$ .

$\frac{1 + \sqrt{13}}{3}, \frac{1 - \sqrt{13}}{3}$

Step 4

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

$(- \infty, \frac{1 - \sqrt{13}}{3}) \cup (\frac{1 - \sqrt{13}}{3}, \frac{1 + \sqrt{13}}{3}) \cup (\frac{1 + \sqrt{13}}{3}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, \frac{1 - \sqrt{13}}{3})$ into the derivative to determine if the function is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 1.8685171$ in the expression.

$f' (- 1.8685171) = \frac{3 {(- 1.8685171)}^{2}}{2} - (- 1.8685171) - 2$

Step 5.2

Simplify the result.

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Step 5.2.1

Find the common denominator.

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Step 5.2.1.1

Write $- (- 1.8685171)$ as a fraction with denominator $1$ .

$f' (- 1.8685171) = \frac{3 {(- 1.8685171)}^{2}}{2} + \frac{- (- 1.8685171)}{1} - 2$

Step 5.2.1.2

Multiply $\frac{- (- 1.8685171)}{1}$ by $\frac{2}{2}$ .

$f' (- 1.8685171) = \frac{3 {(- 1.8685171)}^{2}}{2} + \frac{- (- 1.8685171)}{1} \cdot \frac{2}{2} - 2$

Step 5.2.1.3

Multiply $\frac{- (- 1.8685171)}{1}$ by $\frac{2}{2}$ .

$f' (- 1.8685171) = \frac{3 {(- 1.8685171)}^{2}}{2} + \frac{1.8685171 \cdot 2}{2} - 2$

Step 5.2.1.4

Write $- 2$ as a fraction with denominator $1$ .

$f' (- 1.8685171) = \frac{3 {(- 1.8685171)}^{2}}{2} + \frac{1.8685171 \cdot 2}{2} + \frac{- 2}{1}$

Step 5.2.1.5

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

$f' (- 1.8685171) = \frac{3 {(- 1.8685171)}^{2}}{2} + \frac{1.8685171 \cdot 2}{2} + \frac{- 2}{1} \cdot \frac{2}{2}$

Step 5.2.1.6

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

$f' (- 1.8685171) = \frac{3 {(- 1.8685171)}^{2}}{2} + \frac{1.8685171 \cdot 2}{2} + \frac{- 2 \cdot 2}{2}$

Step 5.2.2

Combine the numerators over the common denominator.

$f' (- 1.8685171) = \frac{3 {(- 1.8685171)}^{2} + 1.8685171 \cdot 2 - 2 \cdot 2}{2}$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Raise $- 1.8685171$ to the power of $2$ .

$f' (- 1.8685171) = \frac{3 \cdot 3.49135615 + 1.8685171 \cdot 2 - 2 \cdot 2}{2}$

Step 5.2.3.2

Multiply $3$ by $3.49135615$ .

$f' (- 1.8685171) = \frac{10.47406845 + 1.8685171 \cdot 2 - 2 \cdot 2}{2}$

Step 5.2.3.3

Multiply $- (- 1.8685171) \cdot 2$ .

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Step 5.2.3.3.1

Multiply $- 1$ by $- 1.8685171$ .

$f' (- 1.8685171) = \frac{10.47406845 + 1.8685171 \cdot 2 - 2 \cdot 2}{2}$

Step 5.2.3.3.2

Multiply $1.8685171$ by $2$ .

$f' (- 1.8685171) = \frac{10.47406845 + 3.7370342 - 2 \cdot 2}{2}$

Step 5.2.3.4

Multiply $- 2$ by $2$ .

$f' (- 1.8685171) = \frac{10.47406845 + 3.7370342 - 4}{2}$

Step 5.2.4

Simplify the expression.

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Step 5.2.4.1

Add $10.47406845$ and $3.7370342$ .

$f' (- 1.8685171) = \frac{14.21110265 - 4}{2}$

Step 5.2.4.2

Subtract $4$ from $14.21110265$ .

$f' (- 1.8685171) = \frac{10.21110265}{2}$

Step 5.2.4.3

Divide $10.21110265$ by $2$ .

$f' (- 1.8685171) = 5.10555132$

Step 5.2.5

The final answer is $5.10555132$ .

$5.10555132$

Step 5.3

At $x = - 1.8685171$ the derivative is $5.10555132$ . Since this is positive, the function is increasing on $(- \infty, \frac{1 - \sqrt{13}}{3})$ .

Increasing on $(- \infty, \frac{1 - \sqrt{13}}{3})$ since $f' (x) > 0$

Step 6

Substitute a value from the interval $(- 0.8685171, \frac{1 + \sqrt{13}}{3})$ into the derivative to determine if the function is increasing or decreasing.

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Step 6.1

Replace the variable $x$ with $0.\overline{3}$ in the expression.

$f' (0.\overline{3}) = \frac{3 {(0.\overline{3})}^{2}}{2} - (0.\overline{3}) - 2$

Step 6.2

Simplify the result.

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Step 6.2.1

Find the common denominator.

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Step 6.2.1.1

Write $- (0.\overline{3})$ as a fraction with denominator $1$ .

$f' (0.\overline{3}) = \frac{3 {(0.\overline{3})}^{2}}{2} + \frac{- (0.\overline{3})}{1} - 2$

Step 6.2.1.2

Multiply $\frac{- (0.\overline{3})}{1}$ by $\frac{2}{2}$ .

$f' (0.\overline{3}) = \frac{3 {(0.\overline{3})}^{2}}{2} + \frac{- (0.\overline{3})}{1} \cdot \frac{2}{2} - 2$

Step 6.2.1.3

Multiply $\frac{- (0.\overline{3})}{1}$ by $\frac{2}{2}$ .

$f' (0.\overline{3}) = \frac{3 {(0.\overline{3})}^{2}}{2} + \frac{- 0.\overline{3} \cdot 2}{2} - 2$

Step 6.2.1.4

Write $- 2$ as a fraction with denominator $1$ .

$f' (0.\overline{3}) = \frac{3 {(0.\overline{3})}^{2}}{2} + \frac{- 0.\overline{3} \cdot 2}{2} + \frac{- 2}{1}$

Step 6.2.1.5

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

$f' (0.\overline{3}) = \frac{3 {(0.\overline{3})}^{2}}{2} + \frac{- 0.\overline{3} \cdot 2}{2} + \frac{- 2}{1} \cdot \frac{2}{2}$

Step 6.2.1.6

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

$f' (0.\overline{3}) = \frac{3 {(0.\overline{3})}^{2}}{2} + \frac{- 0.\overline{3} \cdot 2}{2} + \frac{- 2 \cdot 2}{2}$

Step 6.2.2

Combine the numerators over the common denominator.

$f' (0.\overline{3}) = \frac{3 {(0.\overline{3})}^{2} - 0.\overline{3} \cdot 2 - 2 \cdot 2}{2}$

Step 6.2.3

Simplify each term.

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Step 6.2.3.1

Raise $0.\overline{3}$ to the power of $2$ .

$f' (0.\overline{3}) = \frac{3 \cdot 0.11111112 - 0.\overline{3} \cdot 2 - 2 \cdot 2}{2}$

Step 6.2.3.2

Multiply $3$ by $0.11111112$ .

$f' (0.\overline{3}) = \frac{0.33333336 - 0.\overline{3} \cdot 2 - 2 \cdot 2}{2}$

Step 6.2.3.3

Multiply $- (0.\overline{3}) \cdot 2$ .

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Step 6.2.3.3.1

Multiply $- 1$ by $0.\overline{3}$ .

$f' (0.\overline{3}) = \frac{0.33333336 - 0.\overline{3} \cdot 2 - 2 \cdot 2}{2}$

Step 6.2.3.3.2

Multiply $- 0.\overline{3}$ by $2$ .

$f' (0.\overline{3}) = \frac{0.33333336 - 0.6666667 - 2 \cdot 2}{2}$

Step 6.2.3.4

Multiply $- 2$ by $2$ .

$f' (0.\overline{3}) = \frac{0.33333336 - 0.6666667 - 4}{2}$

Step 6.2.4

Simplify the expression.

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Step 6.2.4.1

Subtract $0.6666667$ from $0.33333336$ .

$f' (0.\overline{3}) = \frac{- 0.\overline{3} - 4}{2}$

Step 6.2.4.2

Subtract $4$ from $- 0.\overline{3}$ .

$f' (0.\overline{3}) = \frac{- 4.\overline{3}}{2}$

Step 6.2.4.3

Divide $- 4.\overline{3}$ by $2$ .

$f' (0.\overline{3}) = - 2.1\overline{6}$

Step 6.2.5

The final answer is $- 2.1\overline{6}$ .

$- 2.1\overline{6}$

Step 6.3

At $x = 0.\overline{3}$ the derivative is $- 2.1\overline{6}$ . Since this is negative, the function is decreasing on $(- 0.8685171, \frac{1 + \sqrt{13}}{3})$ .

Decreasing on $(\frac{1 - \sqrt{13}}{3}, \frac{1 + \sqrt{13}}{3})$ since $f' (x) < 0$

Step 7

Substitute a value from the interval $(\frac{1 + \sqrt{13}}{3}, \infty)$ into the derivative to determine if the function is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $2.5351838$ in the expression.

$f' (2.5351838) = \frac{3 {(2.5351838)}^{2}}{2} - (2.5351838) - 2$

Step 7.2

Simplify the result.

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Step 7.2.1

Find the common denominator.

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Step 7.2.1.1

Write $- (2.5351838)$ as a fraction with denominator $1$ .

$f' (2.5351838) = \frac{3 {(2.5351838)}^{2}}{2} + \frac{- (2.5351838)}{1} - 2$

Step 7.2.1.2

Multiply $\frac{- (2.5351838)}{1}$ by $\frac{2}{2}$ .

$f' (2.5351838) = \frac{3 {(2.5351838)}^{2}}{2} + \frac{- (2.5351838)}{1} \cdot \frac{2}{2} - 2$

Step 7.2.1.3

Multiply $\frac{- (2.5351838)}{1}$ by $\frac{2}{2}$ .

$f' (2.5351838) = \frac{3 {(2.5351838)}^{2}}{2} + \frac{- 2.5351838 \cdot 2}{2} - 2$

Step 7.2.1.4

Write $- 2$ as a fraction with denominator $1$ .

$f' (2.5351838) = \frac{3 {(2.5351838)}^{2}}{2} + \frac{- 2.5351838 \cdot 2}{2} + \frac{- 2}{1}$

Step 7.2.1.5

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

$f' (2.5351838) = \frac{3 {(2.5351838)}^{2}}{2} + \frac{- 2.5351838 \cdot 2}{2} + \frac{- 2}{1} \cdot \frac{2}{2}$

Step 7.2.1.6

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

$f' (2.5351838) = \frac{3 {(2.5351838)}^{2}}{2} + \frac{- 2.5351838 \cdot 2}{2} + \frac{- 2 \cdot 2}{2}$

Step 7.2.2

Combine the numerators over the common denominator.

$f' (2.5351838) = \frac{3 {(2.5351838)}^{2} - 2.5351838 \cdot 2 - 2 \cdot 2}{2}$

Step 7.2.3

Simplify each term.

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Step 7.2.3.1

Raise $2.5351838$ to the power of $2$ .

$f' (2.5351838) = \frac{3 \cdot 6.42715689 - 2.5351838 \cdot 2 - 2 \cdot 2}{2}$

Step 7.2.3.2

Multiply $3$ by $6.42715689$ .

$f' (2.5351838) = \frac{19.28147069 - 2.5351838 \cdot 2 - 2 \cdot 2}{2}$

Step 7.2.3.3

Multiply $- (2.5351838) \cdot 2$ .

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Step 7.2.3.3.1

Multiply $- 1$ by $2.5351838$ .

$f' (2.5351838) = \frac{19.28147069 - 2.5351838 \cdot 2 - 2 \cdot 2}{2}$

Step 7.2.3.3.2

Multiply $- 2.5351838$ by $2$ .

$f' (2.5351838) = \frac{19.28147069 - 5.0703676 - 2 \cdot 2}{2}$

Step 7.2.3.4

Multiply $- 2$ by $2$ .

$f' (2.5351838) = \frac{19.28147069 - 5.0703676 - 4}{2}$

Step 7.2.4

Simplify the expression.

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Step 7.2.4.1

Subtract $5.0703676$ from $19.28147069$ .

$f' (2.5351838) = \frac{14.21110309 - 4}{2}$

Step 7.2.4.2

Subtract $4$ from $14.21110309$ .

$f' (2.5351838) = \frac{10.21110309}{2}$

Step 7.2.4.3

Divide $10.21110309$ by $2$ .

$f' (2.5351838) = 5.10555154$

Step 7.2.5

The final answer is $5.10555154$ .

$5.10555154$

Step 7.3

At $x = 2.5351838$ the derivative is $5.10555154$ . Since this is positive, the function is increasing on $(\frac{1 + \sqrt{13}}{3}, \infty)$ .

Increasing on $(\frac{1 + \sqrt{13}}{3}, \infty)$ since $f' (x) > 0$

Step 8

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

Increasing on: $(- \infty, \frac{1 - \sqrt{13}}{3}), (\frac{1 + \sqrt{13}}{3}, \infty)$

Decreasing on: $(\frac{1 - \sqrt{13}}{3}, \frac{1 + \sqrt{13}}{3})$

Step 9