Calculus Examples

$y = \frac{1}{8} x^{4} - 3 x^{2}$

Step 1

Write $y = \frac{1}{8} x^{4} - 3 x^{2}$ as a function.

$f (x) = \frac{1}{8} x^{4} - 3 x^{2}$

Step 2

Find the second derivative.

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Step 2.1

Find the first derivative.

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Step 2.1.1

By the Sum Rule, the derivative of $\frac{1}{8} x^{4} - 3 x^{2}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{8} x^{4}] + \frac{d}{d x} [- 3 x^{2}]$ .

$\frac{d}{d x} [\frac{1}{8} x^{4}] + \frac{d}{d x} [- 3 x^{2}]$

Step 2.1.2

Evaluate $\frac{d}{d x} [\frac{1}{8} x^{4}]$ .

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Step 2.1.2.1

Since $\frac{1}{8}$ is constant with respect to $x$ , the derivative of $\frac{1}{8} x^{4}$ with respect to $x$ is $\frac{1}{8} \frac{d}{d x} [x^{4}]$ .

$\frac{1}{8} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 3 x^{2}]$

Step 2.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{1}{8} (4 x^{3}) + \frac{d}{d x} [- 3 x^{2}]$

Step 2.1.2.3

Combine $4$ and $\frac{1}{8}$ .

$\frac{4}{8} x^{3} + \frac{d}{d x} [- 3 x^{2}]$

Step 2.1.2.4

Combine $\frac{4}{8}$ and $x^{3}$ .

$\frac{4 x^{3}}{8} + \frac{d}{d x} [- 3 x^{2}]$

Step 2.1.2.5

Cancel the common factor of $4$ and $8$ .

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Step 2.1.2.5.1

Factor $4$ out of $4 x^{3}$ .

$\frac{4 (x^{3})}{8} + \frac{d}{d x} [- 3 x^{2}]$

Step 2.1.2.5.2

Cancel the common factors.

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Step 2.1.2.5.2.1

Factor $4$ out of $8$ .

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

Step 2.1.2.5.2.2

Cancel the common factor.

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

Step 2.1.2.5.2.3

Rewrite the expression.

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

Step 2.1.3

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

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Step 2.1.3.1

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

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

Step 2.1.3.2

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

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

Step 2.1.3.3

Multiply $2$ by $- 3$ .

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

Step 2.2

Find the second derivative.

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Step 2.2.1

By the Sum Rule, the derivative of $\frac{x^{3}}{2} - 6 x$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{3}}{2}] + \frac{d}{d x} [- 6 x]$ .

$\frac{d}{d x} [\frac{x^{3}}{2}] + \frac{d}{d x} [- 6 x]$

Step 2.2.2

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

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Step 2.2.2.1

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

$\frac{1}{2} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 6 x]$

Step 2.2.2.2

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

$\frac{1}{2} (3 x^{2}) + \frac{d}{d x} [- 6 x]$

Step 2.2.2.3

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

$\frac{3}{2} x^{2} + \frac{d}{d x} [- 6 x]$

Step 2.2.2.4

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

$\frac{3 x^{2}}{2} + \frac{d}{d x} [- 6 x]$

Step 2.2.3

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

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Step 2.2.3.1

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$\frac{3 x^{2}}{2} - 6 \frac{d}{d x} [x]$

Step 2.2.3.2

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

$\frac{3 x^{2}}{2} - 6 \cdot 1$

Step 2.2.3.3

Multiply $- 6$ by $1$ .

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

Step 2.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{3 x^{2}}{2} - 6$ .

$\frac{3 x^{2}}{2} - 6$

Step 3

Set the second derivative equal to $0$ then solve the equation $\frac{3 x^{2}}{2} - 6 = 0$ .

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Step 3.1

Set the second derivative equal to $0$ .

$\frac{3 x^{2}}{2} - 6 = 0$

Step 3.2

Add $6$ to both sides of the equation.

$\frac{3 x^{2}}{2} = 6$

Step 3.3

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} \cdot \frac{3 x^{2}}{2} = \frac{2}{3} \cdot 6$

Step 3.4

Simplify both sides of the equation.

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Step 3.4.1

Simplify the left side.

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Step 3.4.1.1

Simplify $\frac{2}{3} \cdot \frac{3 x^{2}}{2}$ .

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Step 3.4.1.1.1

Combine.

$\frac{2 (3 x^{2})}{3 \cdot 2} = \frac{2}{3} \cdot 6$

Step 3.4.1.1.2

Cancel the common factor of $2$ .

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Step 3.4.1.1.2.1

Cancel the common factor.

$\frac{2 (3 x^{2})}{3 \cdot 2} = \frac{2}{3} \cdot 6$

Step 3.4.1.1.2.2

Rewrite the expression.

$\frac{3 x^{2}}{3} = \frac{2}{3} \cdot 6$

Step 3.4.1.1.3

Cancel the common factor of $3$ .

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Step 3.4.1.1.3.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{2}{3} \cdot 6$

Step 3.4.1.1.3.2

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

$x^{2} = \frac{2}{3} \cdot 6$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $\frac{2}{3} \cdot 6$ .

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Step 3.4.2.1.1

Cancel the common factor of $3$ .

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Step 3.4.2.1.1.1

Factor $3$ out of $6$ .

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

Step 3.4.2.1.1.2

Cancel the common factor.

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

Step 3.4.2.1.1.3

Rewrite the expression.

$x^{2} = 2 \cdot 2$

Step 3.4.2.1.2

Multiply $2$ by $2$ .

$x^{2} = 4$

Step 3.5

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

$x = \pm \sqrt{4}$

Step 3.6

Simplify $\pm \sqrt{4}$ .

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Step 3.6.1

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

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

Step 3.6.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

Step 3.7

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

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Step 3.7.1

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

$x = 2$

Step 3.7.2

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

$x = - 2$

Step 3.7.3

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

$x = 2, - 2$

Step 4

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

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Step 4.1

Substitute $2$ in $f (x) = \frac{1}{8} x^{4} - 3 x^{2}$ to find the value of $y$ .

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Step 4.1.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = \frac{1}{8} \cdot {(2)}^{4} - 3 {(2)}^{2}$

Step 4.1.2

Simplify the result.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Raise $2$ to the power of $4$ .

$f (2) = \frac{1}{8} \cdot 16 - 3 {(2)}^{2}$

Step 4.1.2.1.2

Cancel the common factor of $8$ .

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Step 4.1.2.1.2.1

Factor $8$ out of $16$ .

$f (2) = \frac{1}{8} \cdot (8 (2)) - 3 {(2)}^{2}$

Step 4.1.2.1.2.2

Cancel the common factor.

$f (2) = \frac{1}{8} \cdot (8 \cdot 2) - 3 {(2)}^{2}$

Step 4.1.2.1.2.3

Rewrite the expression.

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

Step 4.1.2.1.3

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

$f (2) = 2 - 3 \cdot 4$

Step 4.1.2.1.4

Multiply $- 3$ by $4$ .

$f (2) = 2 - 12$

Step 4.1.2.2

Subtract $12$ from $2$ .

$f (2) = - 10$

Step 4.1.2.3

The final answer is $- 10$ .

$- 10$

Step 4.2

The point found by substituting $2$ in $f (x) = \frac{1}{8} x^{4} - 3 x^{2}$ is $(2, - 10)$ . This point can be an inflection point.

$(2, - 10)$

Step 4.3

Substitute $- 2$ in $f (x) = \frac{1}{8} x^{4} - 3 x^{2}$ to find the value of $y$ .

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Step 4.3.1

Replace the variable $x$ with $- 2$ in the expression.

$f (- 2) = \frac{1}{8} \cdot {(- 2)}^{4} - 3 {(- 2)}^{2}$

Step 4.3.2

Simplify the result.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Raise $- 2$ to the power of $4$ .

$f (- 2) = \frac{1}{8} \cdot 16 - 3 {(- 2)}^{2}$

Step 4.3.2.1.2

Cancel the common factor of $8$ .

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Step 4.3.2.1.2.1

Factor $8$ out of $16$ .

$f (- 2) = \frac{1}{8} \cdot (8 (2)) - 3 {(- 2)}^{2}$

Step 4.3.2.1.2.2

Cancel the common factor.

$f (- 2) = \frac{1}{8} \cdot (8 \cdot 2) - 3 {(- 2)}^{2}$

Step 4.3.2.1.2.3

Rewrite the expression.

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

Step 4.3.2.1.3

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

$f (- 2) = 2 - 3 \cdot 4$

Step 4.3.2.1.4

Multiply $- 3$ by $4$ .

$f (- 2) = 2 - 12$

Step 4.3.2.2

Subtract $12$ from $2$ .

$f (- 2) = - 10$

Step 4.3.2.3

The final answer is $- 10$ .

$- 10$

Step 4.4

The point found by substituting $- 2$ in $f (x) = \frac{1}{8} x^{4} - 3 x^{2}$ is $(- 2, - 10)$ . This point can be an inflection point.

$(- 2, - 10)$

Step 4.5

Determine the points that could be inflection points.

$(2, - 10), (- 2, - 10)$

Step 5

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

$(- \infty, - 2) \cup (- 2, 2) \cup (2, \infty)$

Step 6

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

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Step 6.1

Replace the variable $x$ with $- 2.1$ in the expression.

$f'' (- 2.1) = \frac{3 {(- 2.1)}^{2}}{2} - 6$

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

Raise $- 2.1$ to the power of $2$ .

$f'' (- 2.1) = \frac{3 \cdot 4.41}{2} - 6$

Step 6.2.1.2

Multiply $3$ by $4.41$ .

$f'' (- 2.1) = \frac{13.23}{2} - 6$

Step 6.2.1.3

Divide $13.23$ by $2$ .

$f'' (- 2.1) = 6.615 - 6$

Step 6.2.2

Subtract $6$ from $6.615$ .

$f'' (- 2.1) = 0.615$

Step 6.2.3

The final answer is $0.615$ .

$0.615$

Step 6.3

At $- 2.1$ , the second derivative is $0.615$ . Since this is positive, the second derivative is increasing on the interval $(- \infty, - 2)$ .

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

Step 7

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

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Step 7.1

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

$f'' (0) = \frac{3 {(0)}^{2}}{2} - 6$

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

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

$f'' (0) = \frac{3 \cdot 0}{2} - 6$

Step 7.2.1.2

Multiply $3$ by $0$ .

$f'' (0) = \frac{0}{2} - 6$

Step 7.2.1.3

Divide $0$ by $2$ .

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

Step 7.2.2

Subtract $6$ from $0$ .

$f'' (0) = - 6$

Step 7.2.3

The final answer is $- 6$ .

$- 6$

Step 7.3

At $0$ , the second derivative is $- 6$ . Since this is negative, the second derivative is decreasing on the interval $(- 2, 2)$

Decreasing on $(- 2, 2)$ since $f'' (x) < 0$

Step 8

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

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Step 8.1

Replace the variable $x$ with $2.1$ in the expression.

$f'' (2.1) = \frac{3 {(2.1)}^{2}}{2} - 6$

Step 8.2

Simplify the result.

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Step 8.2.1

Simplify each term.

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Step 8.2.1.1

Raise $2.1$ to the power of $2$ .

$f'' (2.1) = \frac{3 \cdot 4.41}{2} - 6$

Step 8.2.1.2

Multiply $3$ by $4.41$ .

$f'' (2.1) = \frac{13.23}{2} - 6$

Step 8.2.1.3

Divide $13.23$ by $2$ .

$f'' (2.1) = 6.615 - 6$

Step 8.2.2

Subtract $6$ from $6.615$ .

$f'' (2.1) = 0.615$

Step 8.2.3

The final answer is $0.615$ .

$0.615$

Step 8.3

At $2.1$ , the second derivative is $0.615$ . Since this is positive, the second derivative is increasing on the interval $(2, \infty)$ .

Increasing on $(2, \infty)$ since $f'' (x) > 0$

Step 9

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 $(- 2, - 10), (2, - 10)$ .

$(- 2, - 10), (2, - 10)$

Step 10