Calculus Examples

$g (x) = 2 x^{4} + 12 x^{2} - 10$

Step 1

Find the second 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 $2 x^{4} + 12 x^{2} - 10$ with respect to $x$ is $\frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 10]$ .

$\frac{d}{d x} [2 x^{4}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 10]$

Step 1.1.2

Evaluate $\frac{d}{d x} [2 x^{4}]$ .

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Step 1.1.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x^{4}$ with respect to $x$ is $2 \frac{d}{d x} [x^{4}]$ .

$2 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 10]$

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 $n = 4$ .

$2 (4 x^{3}) + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 10]$

Step 1.1.2.3

Multiply $4$ by $2$ .

$8 x^{3} + \frac{d}{d x} [12 x^{2}] + \frac{d}{d x} [- 10]$

Step 1.1.3

Evaluate $\frac{d}{d x} [12 x^{2}]$ .

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Step 1.1.3.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{2}$ with respect to $x$ is $12 \frac{d}{d x} [x^{2}]$ .

$8 x^{3} + 12 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 10]$

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 $n = 2$ .

$8 x^{3} + 12 (2 x) + \frac{d}{d x} [- 10]$

Step 1.1.3.3

Multiply $2$ by $12$ .

$8 x^{3} + 24 x + \frac{d}{d x} [- 10]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $- 10$ is constant with respect to $x$ , the derivative of $- 10$ with respect to $x$ is $0$ .

$8 x^{3} + 24 x + 0$

Step 1.1.4.2

Add $8 x^{3} + 24 x$ and $0$ .

$f' (x) = 8 x^{3} + 24 x$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $8 x^{3} + 24 x$ with respect to $x$ is $\frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [24 x]$ .

$\frac{d}{d x} [8 x^{3}] + \frac{d}{d x} [24 x]$

Step 1.2.2

Evaluate $\frac{d}{d x} [8 x^{3}]$ .

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Step 1.2.2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{3}$ with respect to $x$ is $8 \frac{d}{d x} [x^{3}]$ .

$8 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [24 x]$

Step 1.2.2.2

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

$8 (3 x^{2}) + \frac{d}{d x} [24 x]$

Step 1.2.2.3

Multiply $3$ by $8$ .

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

Step 1.2.3

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

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Step 1.2.3.1

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

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

Step 1.2.3.2

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

$24 x^{2} + 24 \cdot 1$

Step 1.2.3.3

Multiply $24$ by $1$ .

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

Step 1.3

The second derivative of $g (x)$ with respect to $x$ is $24 x^{2} + 24$ .

$24 x^{2} + 24$

Step 2

Set the second derivative equal to $0$ then solve the equation $24 x^{2} + 24 = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$24 x^{2} + 24 = 0$

Step 2.2

Subtract $24$ from both sides of the equation.

$24 x^{2} = - 24$

Step 2.3

Divide each term in $24 x^{2} = - 24$ by $24$ and simplify.

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Step 2.3.1

Divide each term in $24 x^{2} = - 24$ by $24$ .

$\frac{24 x^{2}}{24} = \frac{- 24}{24}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $24$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{24 x^{2}}{24} = \frac{- 24}{24}$

Step 2.3.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 24}{24}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $- 24$ by $24$ .

$x^{2} = - 1$

Step 2.4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 1}$

Step 2.5

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

Step 2.6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 2.6.1

First, use the positive value of the $\pm$ to find the first solution.

$x = i$

Step 2.6.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - i$

Step 2.6.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = i, - i$

Step 3

No values found that can make the second derivative equal to $0$ .

No Inflection Points