Calculus Examples

$f (x) = x^{12} + 5 x^{2}$

Step 1

Find the $x$ values where the second derivative is equal to $0$ .

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Step 1.1

Find the second derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

Differentiate.

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Step 1.1.1.1.1

By the Sum Rule, the derivative of $x^{12} + 5 x^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{12}] + \frac{d}{d x} [5 x^{2}]$ .

$\frac{d}{d x} [x^{12}] + \frac{d}{d x} [5 x^{2}]$

Step 1.1.1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 12$ .

$12 x^{11} + \frac{d}{d x} [5 x^{2}]$

Step 1.1.1.2

Evaluate $\frac{d}{d x} [5 x^{2}]$ .

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Step 1.1.1.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

$12 x^{11} + 5 \frac{d}{d x} [x^{2}]$

Step 1.1.1.2.2

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

$12 x^{11} + 5 (2 x)$

Step 1.1.1.2.3

Multiply $2$ by $5$ .

$f' (x) = 12 x^{11} + 10 x$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of $12 x^{11} + 10 x$ with respect to $x$ is $\frac{d}{d x} [12 x^{11}] + \frac{d}{d x} [10 x]$ .

$\frac{d}{d x} [12 x^{11}] + \frac{d}{d x} [10 x]$

Step 1.1.2.2

Evaluate $\frac{d}{d x} [12 x^{11}]$ .

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Step 1.1.2.2.1

Since $12$ is constant with respect to $x$ , the derivative of $12 x^{11}$ with respect to $x$ is $12 \frac{d}{d x} [x^{11}]$ .

$12 \frac{d}{d x} [x^{11}] + \frac{d}{d x} [10 x]$

Step 1.1.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 11$ .

$12 (11 x^{10}) + \frac{d}{d x} [10 x]$

Step 1.1.2.2.3

Multiply $11$ by $12$ .

$132 x^{10} + \frac{d}{d x} [10 x]$

Step 1.1.2.3

Evaluate $\frac{d}{d x} [10 x]$ .

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Step 1.1.2.3.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x$ with respect to $x$ is $10 \frac{d}{d x} [x]$ .

$132 x^{10} + 10 \frac{d}{d x} [x]$

Step 1.1.2.3.2

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

$132 x^{10} + 10 \cdot 1$

Step 1.1.2.3.3

Multiply $10$ by $1$ .

$f'' (x) = 132 x^{10} + 10$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $132 x^{10} + 10$ .

$132 x^{10} + 10$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $132 x^{10} + 10 = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$132 x^{10} + 10 = 0$

Step 1.2.2

Subtract $10$ from both sides of the equation.

$132 x^{10} = - 10$

Step 1.2.3

Divide each term in $132 x^{10} = - 10$ by $132$ and simplify.

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Step 1.2.3.1

Divide each term in $132 x^{10} = - 10$ by $132$ .

$\frac{132 x^{10}}{132} = \frac{- 10}{132}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $132$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{132 x^{10}}{132} = \frac{- 10}{132}$

Step 1.2.3.2.1.2

Divide $x^{10}$ by $1$ .

$x^{10} = \frac{- 10}{132}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Cancel the common factor of $- 10$ and $132$ .

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Step 1.2.3.3.1.1

Factor $2$ out of $- 10$ .

$x^{10} = \frac{2 (- 5)}{132}$

Step 1.2.3.3.1.2

Cancel the common factors.

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Step 1.2.3.3.1.2.1

Factor $2$ out of $132$ .

$x^{10} = \frac{2 \cdot - 5}{2 \cdot 66}$

Step 1.2.3.3.1.2.2

Cancel the common factor.

$x^{10} = \frac{2 \cdot - 5}{2 \cdot 66}$

Step 1.2.3.3.1.2.3

Rewrite the expression.

$x^{10} = \frac{- 5}{66}$

Step 1.2.3.3.2

Move the negative in front of the fraction.

$x^{10} = - \frac{5}{66}$

Step 1.2.4

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

$x = \pm \sqrt[10]{- \frac{5}{66}}$

Step 1.2.5

Simplify $\pm \sqrt[10]{- \frac{5}{66}}$ .

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Step 1.2.5.1

Rewrite $- 1$ as $i^{10}$ .

$x = \pm \sqrt[10]{i^{10} \frac{5}{66}}$

Step 1.2.5.2

Pull terms out from under the radical.

$x = \pm i \sqrt[10]{\frac{5}{66}}$

Step 1.2.5.3

Rewrite $\sqrt[10]{\frac{5}{66}}$ as $\frac{\sqrt[10]{5}}{\sqrt[10]{66}}$ .

$x = \pm i \frac{\sqrt[10]{5}}{\sqrt[10]{66}}$

Step 1.2.5.4

Combine $i$ and $\frac{\sqrt[10]{5}}{\sqrt[10]{66}}$ .

$x = \pm \frac{i \sqrt[10]{5}}{\sqrt[10]{66}}$

Step 1.2.6

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

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Step 1.2.6.1

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

$x = \frac{i \sqrt[10]{5}}{\sqrt[10]{66}}$

Step 1.2.6.2

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

$x = - \frac{i \sqrt[10]{5}}{\sqrt[10]{66}}$

Step 1.2.6.3

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

$x = \frac{i \sqrt[10]{5}}{\sqrt[10]{66}}, - \frac{i \sqrt[10]{5}}{\sqrt[10]{66}}$

Step 2

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, \infty)$

Step 4

Substitute any number from the interval $(- \infty, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

Replace the variable $x$ with $0$ in the expression.

$f'' (0) = 132 {(0)}^{10} + 10$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raising $0$ to any positive power yields $0$ .

$f'' (0) = 132 \cdot 0 + 10$

Step 4.2.1.2

Multiply $132$ by $0$ .

$f'' (0) = 0 + 10$

Step 4.2.2

Add $0$ and $10$ .

$f'' (0) = 10$

Step 4.2.3

The final answer is $10$ .

$10$

Step 4.3

The graph is concave up on the interval $(- \infty, \infty)$ because $f'' (0)$ is positive.

The graph is concave up

Step 5