Calculus Examples

$f (x) = x^{4} - 12 x^{2}$

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

$f' (x) = \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 12 x^{2})$

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 $n = 4$ .

$f' (x) = 4 x^{3} + \frac{d}{d x} (- 12 x^{2})$

Step 1.1.2

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

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Step 1.1.2.1

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

$f' (x) = 4 x^{3} - 12 \frac{d}{d x} x^{2}$

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 $n = 2$ .

$f' (x) = 4 x^{3} - 12 (2 x)$

Step 1.1.2.3

Multiply $2$ by $- 12$ .

$f' (x) = 4 x^{3} - 24 x$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $4 x^{3} - 24 x$ with respect to $x$ is $\frac{d}{d x} [4 x^{3}] + \frac{d}{d x} [- 24 x]$ .

$f'' (x) = \frac{d}{d x} (4 x^{3}) + \frac{d}{d x} (- 24 x)$

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

$f'' (x) = 4 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 24 x)$

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 $n = 3$ .

$f'' (x) = 4 (3 x^{2}) + \frac{d}{d x} (- 24 x)$

Step 1.2.2.3

Multiply $3$ by $4$ .

$f'' (x) = 12 x^{2} + \frac{d}{d x} (- 24 x)$

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

$f'' (x) = 12 x^{2} - 24 \frac{d}{d x} x$

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 $n = 1$ .

$f'' (x) = 12 x^{2} - 24 \cdot 1$

Step 1.2.3.3

Multiply $- 24$ by $1$ .

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

Step 1.3

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

$12 x^{2} - 24$

Step 2

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

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Step 2.1

Set the second derivative equal to $0$ .

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

Step 2.2

Add $24$ to both sides of the equation.

$12 x^{2} = 24$

Step 2.3

Divide each term in $12 x^{2} = 24$ by $12$ and simplify.

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Step 2.3.1

Divide each term in $12 x^{2} = 24$ by $12$ .

$\frac{12 x^{2}}{12} = \frac{24}{12}$

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.

$\frac{12 x^{2}}{12} = \frac{24}{12}$

Step 2.3.2.1.2

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

$x^{2} = \frac{24}{12}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $24$ by $12$ .

$x^{2} = 2$

Step 2.4

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

$x = \pm \sqrt{2}$

Step 2.5

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

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Step 2.5.1

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

$x = \sqrt{2}$

Step 2.5.2

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

$x = - \sqrt{2}$

Step 2.5.3

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

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

Step 3

Find the points where the second derivative is $0$ .

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Step 3.1

Substitute $\sqrt{2}$ in $f (x) = x^{4} - 12 x^{2}$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $\sqrt{2}$ in the expression.

$f (\sqrt{2}) = {(\sqrt{2})}^{4} - 12 {(\sqrt{2})}^{2}$

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

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

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Step 3.1.2.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$f (\sqrt{2}) = {(2^{\frac{1}{2}})}^{4} - 12 {(\sqrt{2})}^{2}$

Step 3.1.2.1.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\sqrt{2}) = 2^{\frac{1}{2} \cdot 4} - 12 {(\sqrt{2})}^{2}$

Step 3.1.2.1.1.3

Combine $\frac{1}{2}$ and $4$ .

$f (\sqrt{2}) = 2^{\frac{4}{2}} - 12 {(\sqrt{2})}^{2}$

Step 3.1.2.1.1.4

Cancel the common factor of $4$ and $2$ .

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Step 3.1.2.1.1.4.1

Factor $2$ out of $4$ .

$f (\sqrt{2}) = 2^{\frac{2 \cdot 2}{2}} - 12 {(\sqrt{2})}^{2}$

Step 3.1.2.1.1.4.2

Cancel the common factors.

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Step 3.1.2.1.1.4.2.1

Factor $2$ out of $2$ .

$f (\sqrt{2}) = 2^{\frac{2 \cdot 2}{2 (1)}} - 12 {(\sqrt{2})}^{2}$

Step 3.1.2.1.1.4.2.2

Cancel the common factor.

$f (\sqrt{2}) = 2^{\frac{2 \cdot 2}{2 \cdot 1}} - 12 {(\sqrt{2})}^{2}$

Step 3.1.2.1.1.4.2.3

Rewrite the expression.

$f (\sqrt{2}) = 2^{\frac{2}{1}} - 12 {(\sqrt{2})}^{2}$

Step 3.1.2.1.1.4.2.4

Divide $2$ by $1$ .

$f (\sqrt{2}) = 2^{2} - 12 {(\sqrt{2})}^{2}$

Step 3.1.2.1.2

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

$f (\sqrt{2}) = 4 - 12 {(\sqrt{2})}^{2}$

Step 3.1.2.1.3

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

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Step 3.1.2.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$f (\sqrt{2}) = 4 - 12 {(2^{\frac{1}{2}})}^{2}$

Step 3.1.2.1.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (\sqrt{2}) = 4 - 12 \cdot 2^{\frac{1}{2} \cdot 2}$

Step 3.1.2.1.3.3

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

$f (\sqrt{2}) = 4 - 12 \cdot 2^{\frac{2}{2}}$

Step 3.1.2.1.3.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.3.4.1

Cancel the common factor.

$f (\sqrt{2}) = 4 - 12 \cdot 2^{\frac{2}{2}}$

Step 3.1.2.1.3.4.2

Rewrite the expression.

$f (\sqrt{2}) = 4 - 12 \cdot 2$

Step 3.1.2.1.3.5

Evaluate the exponent.

$f (\sqrt{2}) = 4 - 12 \cdot 2$

Step 3.1.2.1.4

Multiply $- 12$ by $2$ .

$f (\sqrt{2}) = 4 - 24$

Step 3.1.2.2

Subtract $24$ from $4$ .

$f (\sqrt{2}) = - 20$

Step 3.1.2.3

The final answer is $- 20$ .

$- 20$

Step 3.2

The point found by substituting $\sqrt{2}$ in $f (x) = x^{4} - 12 x^{2}$ is $(\sqrt{2}, - 20)$ . This point can be an inflection point.

$(\sqrt{2}, - 20)$

Step 3.3

Substitute $- \sqrt{2}$ in $f (x) = x^{4} - 12 x^{2}$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $- \sqrt{2}$ in the expression.

$f (- \sqrt{2}) = {(- \sqrt{2})}^{4} - 12 {(- \sqrt{2})}^{2}$

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

Apply the product rule to $- \sqrt{2}$ .

$f (- \sqrt{2}) = {(- 1)}^{4} {\sqrt{2}}^{4} - 12 {(- \sqrt{2})}^{2}$

Step 3.3.2.1.2

Raise $- 1$ to the power of $4$ .

$f (- \sqrt{2}) = 1 {\sqrt{2}}^{4} - 12 {(- \sqrt{2})}^{2}$

Step 3.3.2.1.3

Multiply ${\sqrt{2}}^{4}$ by $1$ .

$f (- \sqrt{2}) = {\sqrt{2}}^{4} - 12 {(- \sqrt{2})}^{2}$

Step 3.3.2.1.4

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

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Step 3.3.2.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$f (- \sqrt{2}) = {(2^{\frac{1}{2}})}^{4} - 12 {(- \sqrt{2})}^{2}$

Step 3.3.2.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \sqrt{2}) = 2^{\frac{1}{2} \cdot 4} - 12 {(- \sqrt{2})}^{2}$

Step 3.3.2.1.4.3

Combine $\frac{1}{2}$ and $4$ .

$f (- \sqrt{2}) = 2^{\frac{4}{2}} - 12 {(- \sqrt{2})}^{2}$

Step 3.3.2.1.4.4

Cancel the common factor of $4$ and $2$ .

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Step 3.3.2.1.4.4.1

Factor $2$ out of $4$ .

$f (- \sqrt{2}) = 2^{\frac{2 \cdot 2}{2}} - 12 {(- \sqrt{2})}^{2}$

Step 3.3.2.1.4.4.2

Cancel the common factors.

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Step 3.3.2.1.4.4.2.1

Factor $2$ out of $2$ .

$f (- \sqrt{2}) = 2^{\frac{2 \cdot 2}{2 (1)}} - 12 {(- \sqrt{2})}^{2}$

Step 3.3.2.1.4.4.2.2

Cancel the common factor.

$f (- \sqrt{2}) = 2^{\frac{2 \cdot 2}{2 \cdot 1}} - 12 {(- \sqrt{2})}^{2}$

Step 3.3.2.1.4.4.2.3

Rewrite the expression.

$f (- \sqrt{2}) = 2^{\frac{2}{1}} - 12 {(- \sqrt{2})}^{2}$

Step 3.3.2.1.4.4.2.4

Divide $2$ by $1$ .

$f (- \sqrt{2}) = 2^{2} - 12 {(- \sqrt{2})}^{2}$

Step 3.3.2.1.5

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

$f (- \sqrt{2}) = 4 - 12 {(- \sqrt{2})}^{2}$

Step 3.3.2.1.6

Apply the product rule to $- \sqrt{2}$ .

$f (- \sqrt{2}) = 4 - 12 ({(- 1)}^{2} {\sqrt{2}}^{2})$

Step 3.3.2.1.7

Raise $- 1$ to the power of $2$ .

$f (- \sqrt{2}) = 4 - 12 (1 {\sqrt{2}}^{2})$

Step 3.3.2.1.8

Multiply ${\sqrt{2}}^{2}$ by $1$ .

$f (- \sqrt{2}) = 4 - 12 {\sqrt{2}}^{2}$

Step 3.3.2.1.9

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

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Step 3.3.2.1.9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$f (- \sqrt{2}) = 4 - 12 {(2^{\frac{1}{2}})}^{2}$

Step 3.3.2.1.9.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f (- \sqrt{2}) = 4 - 12 \cdot 2^{\frac{1}{2} \cdot 2}$

Step 3.3.2.1.9.3

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

$f (- \sqrt{2}) = 4 - 12 \cdot 2^{\frac{2}{2}}$

Step 3.3.2.1.9.4

Cancel the common factor of $2$ .

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Step 3.3.2.1.9.4.1

Cancel the common factor.

$f (- \sqrt{2}) = 4 - 12 \cdot 2^{\frac{2}{2}}$

Step 3.3.2.1.9.4.2

Rewrite the expression.

$f (- \sqrt{2}) = 4 - 12 \cdot 2$

Step 3.3.2.1.9.5

Evaluate the exponent.

$f (- \sqrt{2}) = 4 - 12 \cdot 2$

Step 3.3.2.1.10

Multiply $- 12$ by $2$ .

$f (- \sqrt{2}) = 4 - 24$

Step 3.3.2.2

Subtract $24$ from $4$ .

$f (- \sqrt{2}) = - 20$

Step 3.3.2.3

The final answer is $- 20$ .

$- 20$

Step 3.4

The point found by substituting $- \sqrt{2}$ in $f (x) = x^{4} - 12 x^{2}$ is $(- \sqrt{2}, - 20)$ . This point can be an inflection point.

$(- \sqrt{2}, - 20)$

Step 3.5

Determine the points that could be inflection points.

$(\sqrt{2}, - 20), (- \sqrt{2}, - 20)$

Step 4

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

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

Step 5

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

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Step 5.1

Replace the variable $x$ with $- 1.51421356$ in the expression.

$f'' (- 1.51421356) = 12 {(- 1.51421356)}^{2} - 24$

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 $- 1.51421356$ to the power of $2$ .

$f'' (- 1.51421356) = 12 \cdot 2.29284271 - 24$

Step 5.2.1.2

Multiply $12$ by $2.29284271$ .

$f'' (- 1.51421356) = 27.51411254 - 24$

Step 5.2.2

Subtract $24$ from $27.51411254$ .

$f'' (- 1.51421356) = 3.51411254$

Step 5.2.3

The final answer is $3.51411254$ .

$3.51411254$

Step 5.3

At $- 1.51421356$ , the second derivative is $3.51411254$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - \sqrt{2})$ .

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

Step 6

Substitute a value from the interval $(- \sqrt{2}, \sqrt{2})$ 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.

$f'' (0) = 12 {(0)}^{2} - 24$

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

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

$f'' (0) = 12 \cdot 0 - 24$

Step 6.2.1.2

Multiply $12$ by $0$ .

$f'' (0) = 0 - 24$

Step 6.2.2

Subtract $24$ from $0$ .

$f'' (0) = - 24$

Step 6.2.3

The final answer is $- 24$ .

$- 24$

Step 6.3

At $0$ , the second derivative is $- 24$ . Since this is negative, the second derivative is decreasing on the interval $(- \sqrt{2}, \sqrt{2})$

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

Step 7

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

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Step 7.1

Replace the variable $x$ with $1.51421356$ in the expression.

$f'' (1.51421356) = 12 {(1.51421356)}^{2} - 24$

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 $1.51421356$ to the power of $2$ .

$f'' (1.51421356) = 12 \cdot 2.29284271 - 24$

Step 7.2.1.2

Multiply $12$ by $2.29284271$ .

$f'' (1.51421356) = 27.51411254 - 24$

Step 7.2.2

Subtract $24$ from $27.51411254$ .

$f'' (1.51421356) = 3.51411254$

Step 7.2.3

The final answer is $3.51411254$ .

$3.51411254$

Step 7.3

At $1.51421356$ , the second derivative is $3.51411254$ . Since this is positive, the second derivative is increasing on the interval $(\sqrt{2}, \infty)$ .

Increasing on $(\sqrt{2}, \infty)$ 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 $(- \sqrt{2}, - 20), (\sqrt{2}, - 20)$ .

$(- \sqrt{2}, - 20), (\sqrt{2}, - 20)$

Step 9