Calculus Examples

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

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

Differentiate.

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Step 1.1.1.1

By the Sum Rule, the derivative of

$x^{4} - 3 x^{2}$ with respect to

$x$ is

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 3 x^{2}]$ .

$f' (x) = \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 3 x^{2})$

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

$n = 4$ .

$f' (x) = 4 x^{3} + \frac{d}{d x} (- 3 x^{2})$

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

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

$f' (x) = 4 x^{3} - 3 \frac{d}{d x} x^{2}$

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

$n = 2$ .

$f' (x) = 4 x^{3} - 3 (2 x)$

Step 1.1.2.3

Multiply

$2$ by

$- 3$ .

$f' (x) = 4 x^{3} - 6 x$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of

$4 x^{3} - 6 x$ with respect to

$x$ is

$\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 6 x]$ .

$f'' (x) = \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 6 x)$

Step 1.2.2

Evaluate

$\frac{d}{d x} [4 x^{3}]$ .

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Step 1.2.2.1

Since

$4$ is constant with respect to

$x$ , the derivative of

$4 x^{3}$ with respect to

$x$ is

$4 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 6 x)$

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

$n = 3$ .

$f'' (x) = 4 (3 x^{2}) + \frac{d}{d x} (- 6 x)$

Step 1.2.2.3

Multiply

$3$ by

$4$ .

$f'' (x) = 12 x^{2} + \frac{d}{d x} (- 6 x)$

Step 1.2.3

Evaluate

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

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Step 1.2.3.1

Since

$- 6$ is constant with respect to

$x$ , the derivative of

$- 6 x$ with respect to

$x$ is

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

$f'' (x) = 12 x^{2} - 6 \frac{d}{d x} x$

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

$n = 1$ .

$f'' (x) = 12 x^{2} - 6 \cdot 1$

Step 1.2.3.3

Multiply

$- 6$ by

$1$ .

$f'' (x) = 12 x^{2} - 6$

Step 1.3

The second derivative of

$f (x)$ with respect to

$x$ is

$12 x^{2} - 6$ .

$12 x^{2} - 6$

Step 2

Set the second derivative equal to

$0$ then solve the equation

$12 x^{2} - 6 = 0$ .

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Step 2.1

Set the second derivative equal to

$0$ .

$12 x^{2} - 6 = 0$

Step 2.2

Add

$6$ to both sides of the equation.

$12 x^{2} = 6$

Step 2.3

Divide each term in

$12 x^{2} = 6$ by

$12$ and simplify.

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Step 2.3.1

Divide each term in

$12 x^{2} = 6$ by

$12$ .

$\frac{12 x^{2}}{12} = \frac{6}{12}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of

$12$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{12 x^{2}}{12} = \frac{6}{12}$

Step 2.3.2.1.2

Divide

$x^{2}$ by

$1$ .

$x^{2} = \frac{6}{12}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Cancel the common factor of

$6$ and

$12$ .

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Step 2.3.3.1.1

Factor

$6$ out of

$6$ .

$x^{2} = \frac{6 (1)}{12}$

Step 2.3.3.1.2

Cancel the common factors.

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Step 2.3.3.1.2.1

Factor

$6$ out of

$12$ .

$x^{2} = \frac{6 \cdot 1}{6 \cdot 2}$

Step 2.3.3.1.2.2

Cancel the common factor.

$x^{2} = \frac{6 \cdot 1}{6 \cdot 2}$

Step 2.3.3.1.2.3

Rewrite the expression.

$x^{2} = \frac{1}{2}$

Step 2.4

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

$x = \pm \sqrt{\frac{1}{2}}$

Step 2.5

Simplify

$\pm \sqrt{\frac{1}{2}}$ .

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Step 2.5.1

Rewrite

$\sqrt{\frac{1}{2}}$ as

$\frac{\sqrt{1}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{2}}$

Step 2.5.2

Any root of

$1$ is

$1$ .

$x = \pm \frac{1}{\sqrt{2}}$

Step 2.5.3

Multiply

$\frac{1}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 2.5.4

Combine and simplify the denominator.

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Step 2.5.4.1

Multiply

$\frac{1}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 2.5.4.2

Raise

$\sqrt{2}$ to the power of

$1$ .

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

Step 2.5.4.3

Raise

$\sqrt{2}$ to the power of

$1$ .

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

Step 2.5.4.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 2.5.4.5

Add

$1$ and

$1$ .

$x = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{2}}$

Step 2.5.4.6

Rewrite

${\sqrt{2}}^{2}$ as

$2$ .

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Step 2.5.4.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 2.5.4.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.5.4.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$x = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.5.4.6.4

Cancel the common factor of

$2$ .

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Step 2.5.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{2}}{2^{\frac{2}{2}}}$

Step 2.5.4.6.4.2

Rewrite the expression.

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

Step 2.5.4.6.5

Evaluate the exponent.

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

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.

$x = \frac{\sqrt{2}}{2}$

Step 2.6.2

Next, use the negative value of the

$\pm$ to find the second solution.

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

Step 2.6.3

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

$x = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Step 3

Find the points where the second derivative is

$0$ .

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Step 3.1

Substitute

$\frac{\sqrt{2}}{2}$ in

$f (x) = x^{4} - 3 x^{2}$ to find the value of

$y$ .

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Step 3.1.1

Replace the variable

$x$ with

$\frac{\sqrt{2}}{2}$ in the expression.

$f (\frac{\sqrt{2}}{2}) = {(\frac{\sqrt{2}}{2})}^{4} - 3 {(\frac{\sqrt{2}}{2})}^{2}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Apply the product rule to

$\frac{\sqrt{2}}{2}$ .

$f (\frac{\sqrt{2}}{2}) = \frac{{\sqrt{2}}^{4}}{2^{4}} - 3 {(\frac{\sqrt{2}}{2})}^{2}$

Step 3.1.2.1.2

Simplify the numerator.

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Step 3.1.2.1.2.1

Rewrite

${\sqrt{2}}^{4}$ as

$2^{2}$ .

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Step 3.1.2.1.2.1.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$f (\frac{\sqrt{2}}{2}) = \frac{{(2^{\frac{1}{2}})}^{4}}{2^{4}} - 3 {(\frac{\sqrt{2}}{2})}^{2}$

Step 3.1.2.1.2.1.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{\sqrt{2}}{2}) = \frac{2^{\frac{1}{2} \cdot 4}}{2^{4}} - 3 {(\frac{\sqrt{2}}{2})}^{2}$

Step 3.1.2.1.2.1.3

Combine

$\frac{1}{2}$ and

$4$ .

$f (\frac{\sqrt{2}}{2}) = \frac{2^{\frac{4}{2}}}{2^{4}} - 3 {(\frac{\sqrt{2}}{2})}^{2}$

Step 3.1.2.1.2.1.4

Cancel the common factor of

$4$ and

$2$ .

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Step 3.1.2.1.2.1.4.1

Factor

$2$ out of

$4$ .

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

Step 3.1.2.1.2.1.4.2

Cancel the common factors.

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Step 3.1.2.1.2.1.4.2.1

Factor

$2$ out of

$2$ .

$f (\frac{\sqrt{2}}{2}) = \frac{2^{\frac{2 \cdot 2}{2 (1)}}}{2^{4}} - 3 {(\frac{\sqrt{2}}{2})}^{2}$

Step 3.1.2.1.2.1.4.2.2

Cancel the common factor.

$f (\frac{\sqrt{2}}{2}) = \frac{2^{\frac{2 \cdot 2}{2 \cdot 1}}}{2^{4}} - 3 {(\frac{\sqrt{2}}{2})}^{2}$

Step 3.1.2.1.2.1.4.2.3

Rewrite the expression.

$f (\frac{\sqrt{2}}{2}) = \frac{2^{\frac{2}{1}}}{2^{4}} - 3 {(\frac{\sqrt{2}}{2})}^{2}$

Step 3.1.2.1.2.1.4.2.4

Divide

$2$ by

$1$ .

$f (\frac{\sqrt{2}}{2}) = \frac{2^{2}}{2^{4}} - 3 {(\frac{\sqrt{2}}{2})}^{2}$

Step 3.1.2.1.2.2

Raise

$2$ to the power of

$2$ .

$f (\frac{\sqrt{2}}{2}) = \frac{4}{2^{4}} - 3 {(\frac{\sqrt{2}}{2})}^{2}$

Step 3.1.2.1.3

Raise

$2$ to the power of

$4$ .

$f (\frac{\sqrt{2}}{2}) = \frac{4}{16} - 3 {(\frac{\sqrt{2}}{2})}^{2}$

Step 3.1.2.1.4

Cancel the common factor of

$4$ and

$16$ .

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Step 3.1.2.1.4.1

Factor

$4$ out of

$4$ .

$f (\frac{\sqrt{2}}{2}) = \frac{4 (1)}{16} - 3 {(\frac{\sqrt{2}}{2})}^{2}$

Step 3.1.2.1.4.2

Cancel the common factors.

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Step 3.1.2.1.4.2.1

Factor

$4$ out of

$16$ .

$f (\frac{\sqrt{2}}{2}) = \frac{4 \cdot 1}{4 \cdot 4} - 3 {(\frac{\sqrt{2}}{2})}^{2}$

Step 3.1.2.1.4.2.2

Cancel the common factor.

$f (\frac{\sqrt{2}}{2}) = \frac{4 \cdot 1}{4 \cdot 4} - 3 {(\frac{\sqrt{2}}{2})}^{2}$

Step 3.1.2.1.4.2.3

Rewrite the expression.

$f (\frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 {(\frac{\sqrt{2}}{2})}^{2}$

Step 3.1.2.1.5

Apply the product rule to

$\frac{\sqrt{2}}{2}$ .

$f (\frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 \frac{{\sqrt{2}}^{2}}{2^{2}}$

Step 3.1.2.1.6

Rewrite

${\sqrt{2}}^{2}$ as

$2$ .

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Step 3.1.2.1.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$f (\frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}}$

Step 3.1.2.1.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$f (\frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}}$

Step 3.1.2.1.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$f (\frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 \frac{2^{\frac{2}{2}}}{2^{2}}$

Step 3.1.2.1.6.4

Cancel the common factor of

$2$ .

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Step 3.1.2.1.6.4.1

Cancel the common factor.

$f (\frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 \frac{2^{\frac{2}{2}}}{2^{2}}$

Step 3.1.2.1.6.4.2

Rewrite the expression.

$f (\frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 \frac{2}{2^{2}}$

Step 3.1.2.1.6.5

Evaluate the exponent.

$f (\frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 \frac{2}{2^{2}}$

Step 3.1.2.1.7

Raise

$2$ to the power of

$2$ .

$f (\frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 (\frac{2}{4})$

Step 3.1.2.1.8

Cancel the common factor of

$2$ and

$4$ .

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Step 3.1.2.1.8.1

Factor

$2$ out of

$2$ .

$f (\frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 \frac{2 (1)}{4}$

Step 3.1.2.1.8.2

Cancel the common factors.

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Step 3.1.2.1.8.2.1

Factor

$2$ out of

$4$ .

$f (\frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 \frac{2 \cdot 1}{2 \cdot 2}$

Step 3.1.2.1.8.2.2

Cancel the common factor.

$f (\frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 \frac{2 \cdot 1}{2 \cdot 2}$

Step 3.1.2.1.8.2.3

Rewrite the expression.

$f (\frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 (\frac{1}{2})$

Step 3.1.2.1.9

Combine

$- 3$ and

$\frac{1}{2}$ .

$f (\frac{\sqrt{2}}{2}) = \frac{1}{4} + \frac{- 3}{2}$

Step 3.1.2.1.10

Move the negative in front of the fraction.

$f (\frac{\sqrt{2}}{2}) = \frac{1}{4} - \frac{3}{2}$

Step 3.1.2.2

To write

$- \frac{3}{2}$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$f (\frac{\sqrt{2}}{2}) = \frac{1}{4} - \frac{3}{2} \cdot \frac{2}{2}$

Step 3.1.2.3

Write each expression with a common denominator of

$4$ , by multiplying each by an appropriate factor of

$1$ .

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Step 3.1.2.3.1

Multiply

$\frac{3}{2}$ by

$\frac{2}{2}$ .

$f (\frac{\sqrt{2}}{2}) = \frac{1}{4} - \frac{3 \cdot 2}{2 \cdot 2}$

Step 3.1.2.3.2

Multiply

$2$ by

$2$ .

$f (\frac{\sqrt{2}}{2}) = \frac{1}{4} - \frac{3 \cdot 2}{4}$

Step 3.1.2.4

Combine the numerators over the common denominator.

$f (\frac{\sqrt{2}}{2}) = \frac{1 - 3 \cdot 2}{4}$

Step 3.1.2.5

Simplify the numerator.

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Step 3.1.2.5.1

Multiply

$- 3$ by

$2$ .

$f (\frac{\sqrt{2}}{2}) = \frac{1 - 6}{4}$

Step 3.1.2.5.2

Subtract

$6$ from

$1$ .

$f (\frac{\sqrt{2}}{2}) = \frac{- 5}{4}$

Step 3.1.2.6

Move the negative in front of the fraction.

$f (\frac{\sqrt{2}}{2}) = - \frac{5}{4}$

Step 3.1.2.7

The final answer is

$- \frac{5}{4}$ .

$- \frac{5}{4}$

Step 3.2

The point found by substituting

$\frac{\sqrt{2}}{2}$ in

$f (x) = x^{4} - 3 x^{2}$ is

$(\frac{\sqrt{2}}{2}, - \frac{5}{4})$ . This point can be an inflection point.

$(\frac{\sqrt{2}}{2}, - \frac{5}{4})$

Step 3.3

Substitute

$- \frac{\sqrt{2}}{2}$ in

$f (x) = x^{4} - 3 x^{2}$ to find the value of

$y$ .

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Step 3.3.1

Replace the variable

$x$ with

$- \frac{\sqrt{2}}{2}$ in the expression.

$f (- \frac{\sqrt{2}}{2}) = {(- \frac{\sqrt{2}}{2})}^{4} - 3 {(- \frac{\sqrt{2}}{2})}^{2}$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.3.2.1.1.1

Apply the product rule to

$- \frac{\sqrt{2}}{2}$ .

$f (- \frac{\sqrt{2}}{2}) = {(- 1)}^{4} {(\frac{\sqrt{2}}{2})}^{4} - 3 {(- \frac{\sqrt{2}}{2})}^{2}$

Step 3.3.2.1.1.2

Apply the product rule to

$\frac{\sqrt{2}}{2}$ .

$f (- \frac{\sqrt{2}}{2}) = {(- 1)}^{4} (\frac{{\sqrt{2}}^{4}}{2^{4}}) - 3 {(- \frac{\sqrt{2}}{2})}^{2}$

Step 3.3.2.1.2

Raise

$- 1$ to the power of

$4$ .

$f (- \frac{\sqrt{2}}{2}) = 1 (\frac{{\sqrt{2}}^{4}}{2^{4}}) - 3 {(- \frac{\sqrt{2}}{2})}^{2}$

Step 3.3.2.1.3

Multiply

$\frac{{\sqrt{2}}^{4}}{2^{4}}$ by

$1$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{{\sqrt{2}}^{4}}{2^{4}} - 3 {(- \frac{\sqrt{2}}{2})}^{2}$

Step 3.3.2.1.4

Simplify the numerator.

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Step 3.3.2.1.4.1

Rewrite

${\sqrt{2}}^{4}$ as

$2^{2}$ .

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Step 3.3.2.1.4.1.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{{(2^{\frac{1}{2}})}^{4}}{2^{4}} - 3 {(- \frac{\sqrt{2}}{2})}^{2}$

Step 3.3.2.1.4.1.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{2^{\frac{1}{2} \cdot 4}}{2^{4}} - 3 {(- \frac{\sqrt{2}}{2})}^{2}$

Step 3.3.2.1.4.1.3

Combine

$\frac{1}{2}$ and

$4$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{2^{\frac{4}{2}}}{2^{4}} - 3 {(- \frac{\sqrt{2}}{2})}^{2}$

Step 3.3.2.1.4.1.4

Cancel the common factor of

$4$ and

$2$ .

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Step 3.3.2.1.4.1.4.1

Factor

$2$ out of

$4$ .

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

Step 3.3.2.1.4.1.4.2

Cancel the common factors.

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Step 3.3.2.1.4.1.4.2.1

Factor

$2$ out of

$2$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{2^{\frac{2 \cdot 2}{2 (1)}}}{2^{4}} - 3 {(- \frac{\sqrt{2}}{2})}^{2}$

Step 3.3.2.1.4.1.4.2.2

Cancel the common factor.

$f (- \frac{\sqrt{2}}{2}) = \frac{2^{\frac{2 \cdot 2}{2 \cdot 1}}}{2^{4}} - 3 {(- \frac{\sqrt{2}}{2})}^{2}$

Step 3.3.2.1.4.1.4.2.3

Rewrite the expression.

$f (- \frac{\sqrt{2}}{2}) = \frac{2^{\frac{2}{1}}}{2^{4}} - 3 {(- \frac{\sqrt{2}}{2})}^{2}$

Step 3.3.2.1.4.1.4.2.4

Divide

$2$ by

$1$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{2^{2}}{2^{4}} - 3 {(- \frac{\sqrt{2}}{2})}^{2}$

Step 3.3.2.1.4.2

Raise

$2$ to the power of

$2$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{4}{2^{4}} - 3 {(- \frac{\sqrt{2}}{2})}^{2}$

Step 3.3.2.1.5

Raise

$2$ to the power of

$4$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{4}{16} - 3 {(- \frac{\sqrt{2}}{2})}^{2}$

Step 3.3.2.1.6

Cancel the common factor of

$4$ and

$16$ .

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Step 3.3.2.1.6.1

Factor

$4$ out of

$4$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{4 (1)}{16} - 3 {(- \frac{\sqrt{2}}{2})}^{2}$

Step 3.3.2.1.6.2

Cancel the common factors.

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Step 3.3.2.1.6.2.1

Factor

$4$ out of

$16$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{4 \cdot 1}{4 \cdot 4} - 3 {(- \frac{\sqrt{2}}{2})}^{2}$

Step 3.3.2.1.6.2.2

Cancel the common factor.

$f (- \frac{\sqrt{2}}{2}) = \frac{4 \cdot 1}{4 \cdot 4} - 3 {(- \frac{\sqrt{2}}{2})}^{2}$

Step 3.3.2.1.6.2.3

Rewrite the expression.

$f (- \frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 {(- \frac{\sqrt{2}}{2})}^{2}$

Step 3.3.2.1.7

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.3.2.1.7.1

Apply the product rule to

$- \frac{\sqrt{2}}{2}$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 ({(- 1)}^{2} {(\frac{\sqrt{2}}{2})}^{2})$

Step 3.3.2.1.7.2

Apply the product rule to

$\frac{\sqrt{2}}{2}$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 ({(- 1)}^{2} (\frac{{\sqrt{2}}^{2}}{2^{2}}))$

Step 3.3.2.1.8

Raise

$- 1$ to the power of

$2$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 (1 (\frac{{\sqrt{2}}^{2}}{2^{2}}))$

Step 3.3.2.1.9

Multiply

$\frac{{\sqrt{2}}^{2}}{2^{2}}$ by

$1$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 \frac{{\sqrt{2}}^{2}}{2^{2}}$

Step 3.3.2.1.10

Rewrite

${\sqrt{2}}^{2}$ as

$2$ .

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Step 3.3.2.1.10.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 \frac{{(2^{\frac{1}{2}})}^{2}}{2^{2}}$

Step 3.3.2.1.10.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 \frac{2^{\frac{1}{2} \cdot 2}}{2^{2}}$

Step 3.3.2.1.10.3

Combine

$\frac{1}{2}$ and

$2$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 \frac{2^{\frac{2}{2}}}{2^{2}}$

Step 3.3.2.1.10.4

Cancel the common factor of

$2$ .

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Step 3.3.2.1.10.4.1

Cancel the common factor.

$f (- \frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 \frac{2^{\frac{2}{2}}}{2^{2}}$

Step 3.3.2.1.10.4.2

Rewrite the expression.

$f (- \frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 \frac{2}{2^{2}}$

Step 3.3.2.1.10.5

Evaluate the exponent.

$f (- \frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 \frac{2}{2^{2}}$

Step 3.3.2.1.11

Raise

$2$ to the power of

$2$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 (\frac{2}{4})$

Step 3.3.2.1.12

Cancel the common factor of

$2$ and

$4$ .

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Step 3.3.2.1.12.1

Factor

$2$ out of

$2$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 \frac{2 (1)}{4}$

Step 3.3.2.1.12.2

Cancel the common factors.

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Step 3.3.2.1.12.2.1

Factor

$2$ out of

$4$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 \frac{2 \cdot 1}{2 \cdot 2}$

Step 3.3.2.1.12.2.2

Cancel the common factor.

$f (- \frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 \frac{2 \cdot 1}{2 \cdot 2}$

Step 3.3.2.1.12.2.3

Rewrite the expression.

$f (- \frac{\sqrt{2}}{2}) = \frac{1}{4} - 3 (\frac{1}{2})$

Step 3.3.2.1.13

Combine

$- 3$ and

$\frac{1}{2}$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{1}{4} + \frac{- 3}{2}$

Step 3.3.2.1.14

Move the negative in front of the fraction.

$f (- \frac{\sqrt{2}}{2}) = \frac{1}{4} - \frac{3}{2}$

Step 3.3.2.2

To write

$- \frac{3}{2}$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{1}{4} - \frac{3}{2} \cdot \frac{2}{2}$

Step 3.3.2.3

Write each expression with a common denominator of

$4$ , by multiplying each by an appropriate factor of

$1$ .

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Step 3.3.2.3.1

Multiply

$\frac{3}{2}$ by

$\frac{2}{2}$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{1}{4} - \frac{3 \cdot 2}{2 \cdot 2}$

Step 3.3.2.3.2

Multiply

$2$ by

$2$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{1}{4} - \frac{3 \cdot 2}{4}$

Step 3.3.2.4

Combine the numerators over the common denominator.

$f (- \frac{\sqrt{2}}{2}) = \frac{1 - 3 \cdot 2}{4}$

Step 3.3.2.5

Simplify the numerator.

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Step 3.3.2.5.1

Multiply

$- 3$ by

$2$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{1 - 6}{4}$

Step 3.3.2.5.2

Subtract

$6$ from

$1$ .

$f (- \frac{\sqrt{2}}{2}) = \frac{- 5}{4}$

Step 3.3.2.6

Move the negative in front of the fraction.

$f (- \frac{\sqrt{2}}{2}) = - \frac{5}{4}$

Step 3.3.2.7

The final answer is

$- \frac{5}{4}$ .

$- \frac{5}{4}$

Step 3.4

The point found by substituting

$- \frac{\sqrt{2}}{2}$ in

$f (x) = x^{4} - 3 x^{2}$ is

$(- \frac{\sqrt{2}}{2}, - \frac{5}{4})$ . This point can be an inflection point.

$(- \frac{\sqrt{2}}{2}, - \frac{5}{4})$

Step 3.5

Determine the points that could be inflection points.

$(\frac{\sqrt{2}}{2}, - \frac{5}{4}), (- \frac{\sqrt{2}}{2}, - \frac{5}{4})$

Step 4

Split

$(- \infty, \infty)$ into intervals around the points that could potentially be inflection points.

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

Step 5

Substitute a value from the interval

$(- \infty, - \frac{\sqrt{2}}{2})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable

$x$ with

$- 0.80710678$ in the expression.

$f'' (- 0.80710678) = 12 {(- 0.80710678)}^{2} - 6$

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

$- 0.80710678$ to the power of

$2$ .

$f'' (- 0.80710678) = 12 \cdot 0.65142135 - 6$

Step 5.2.1.2

Multiply

$12$ by

$0.65142135$ .

$f'' (- 0.80710678) = 7.81705627 - 6$

Step 5.2.2

Subtract

$6$ from

$7.81705627$ .

$f'' (- 0.80710678) = 1.81705627$

Step 5.2.3

The final answer is

$1.81705627$ .

$1.81705627$

Step 5.3

$- 0.80710678$ , the second derivative is

$1.81705627$ . Since this is positive, the second derivative is increasing on the interval

$(- \infty, - \frac{\sqrt{2}}{2})$ .

Increasing on

$(- \infty, - \frac{\sqrt{2}}{2})$ since

$f'' (x) > 0$

Increasing on

$(- \infty, - \frac{\sqrt{2}}{2})$ since

$f'' (x) > 0$

Step 6

Substitute a value from the interval

$(- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

Replace the variable

$x$ with

$0$ in the expression.

$f'' (0) = 12 {(0)}^{2} - 6$

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

Raising

$0$ to any positive power yields

$0$ .

$f'' (0) = 12 \cdot 0 - 6$

Step 6.2.1.2

Multiply

$12$ by

$0$ .

$f'' (0) = 0 - 6$

Step 6.2.2

Subtract

$6$ from

$0$ .

$f'' (0) = - 6$

Step 6.2.3

The final answer is

$- 6$ .

$- 6$

Step 6.3

$0$ , the second derivative is

$- 6$ . Since this is negative, the second derivative is decreasing on the interval

$(- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$

Decreasing on

$(- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$ since

$f'' (x) < 0$

Decreasing on

$(- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$ since

$f'' (x) < 0$

Step 7

Substitute a value from the interval

$(\frac{\sqrt{2}}{2}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable

$x$ with

$0.80710678$ in the expression.

$f'' (0.80710678) = 12 {(0.80710678)}^{2} - 6$

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

$0.80710678$ to the power of

$2$ .

$f'' (0.80710678) = 12 \cdot 0.65142135 - 6$

Step 7.2.1.2

Multiply

$12$ by

$0.65142135$ .

$f'' (0.80710678) = 7.81705627 - 6$

Step 7.2.2

Subtract

$6$ from

$7.81705627$ .

$f'' (0.80710678) = 1.81705627$

Step 7.2.3

The final answer is

$1.81705627$ .

$1.81705627$

Step 7.3

$0.80710678$ , the second derivative is

$1.81705627$ . Since this is positive, the second derivative is increasing on the interval

$(\frac{\sqrt{2}}{2}, \infty)$ .

Increasing on

$(\frac{\sqrt{2}}{2}, \infty)$ since

$f'' (x) > 0$

Increasing on

$(\frac{\sqrt{2}}{2}, \infty)$ since

$f'' (x) > 0$

Step 8

An inflection point is a point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The inflection points in this case are

$(- \frac{\sqrt{2}}{2}, - \frac{5}{4}), (\frac{\sqrt{2}}{2}, - \frac{5}{4})$ .

$(- \frac{\sqrt{2}}{2}, - \frac{5}{4}), (\frac{\sqrt{2}}{2}, - \frac{5}{4})$

Step 9

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