Calculus Examples

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

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

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

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 $n = 4$ .

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

Step 1.1.1.2

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

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Step 1.1.1.2.1

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

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

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 $n = 2$ .

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

Step 1.1.1.2.3

Multiply $2$ by $- 2$ .

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

Step 1.1.1.3

Differentiate using the Constant Rule.

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Step 1.1.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$f' (x) = 4 x^{3} - 4 x + 0$

Step 1.1.1.3.2

Add $4 x^{3} - 4 x$ and $0$ .

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

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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Step 1.1.2.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{3}$ with respect to $x$ is $4 \frac{d}{d x} [x^{3}]$ .

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

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 $n = 3$ .

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

Step 1.1.2.2.3

Multiply $3$ by $4$ .

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

Step 1.1.2.3

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

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Step 1.1.2.3.1

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

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

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 $n = 1$ .

$f'' (x) = 12 x^{2} - 4 \cdot 1$

Step 1.1.2.3.3

Multiply $- 4$ by $1$ .

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

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $12 x^{2} - 4$ .

$12 x^{2} - 4$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $12 x^{2} - 4 = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$12 x^{2} - 4 = 0$

Step 1.2.2

Add $4$ to both sides of the equation.

$12 x^{2} = 4$

Step 1.2.3

Divide each term in $12 x^{2} = 4$ by $12$ and simplify.

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Step 1.2.3.1

Divide each term in $12 x^{2} = 4$ by $12$ .

$\frac{12 x^{2}}{12} = \frac{4}{12}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $12$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{12 x^{2}}{12} = \frac{4}{12}$

Step 1.2.3.2.1.2

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

$x^{2} = \frac{4}{12}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Cancel the common factor of $4$ and $12$ .

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Step 1.2.3.3.1.1

Factor $4$ out of $4$ .

$x^{2} = \frac{4 (1)}{12}$

Step 1.2.3.3.1.2

Cancel the common factors.

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Step 1.2.3.3.1.2.1

Factor $4$ out of $12$ .

$x^{2} = \frac{4 \cdot 1}{4 \cdot 3}$

Step 1.2.3.3.1.2.2

Cancel the common factor.

$x^{2} = \frac{4 \cdot 1}{4 \cdot 3}$

Step 1.2.3.3.1.2.3

Rewrite the expression.

$x^{2} = \frac{1}{3}$

Step 1.2.4

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

$x = \pm \sqrt{\frac{1}{3}}$

Step 1.2.5

Simplify $\pm \sqrt{\frac{1}{3}}$ .

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Step 1.2.5.1

Rewrite $\sqrt{\frac{1}{3}}$ as $\frac{\sqrt{1}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{3}}$

Step 1.2.5.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{3}}$

Step 1.2.5.3

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 1.2.5.4

Combine and simplify the denominator.

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Step 1.2.5.4.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 1.2.5.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 1.2.5.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 1.2.5.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 1.2.5.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 1.2.5.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 1.2.5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 1.2.5.4.6.2

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

$x = \pm \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 1.2.5.4.6.3

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

$x = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.2.5.4.6.4

Cancel the common factor of $2$ .

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Step 1.2.5.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 1.2.5.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{3}}{3^{1}}$

Step 1.2.5.4.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{3}}{3}$

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.

$x = \frac{\sqrt{3}}{3}$

Step 1.2.6.2

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

$x = - \frac{\sqrt{3}}{3}$

Step 1.2.6.3

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

$x = \frac{\sqrt{3}}{3}, - \frac{\sqrt{3}}{3}$

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.

$(- \infty, - \frac{\sqrt{3}}{3}) \cup (- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3}) \cup (\frac{\sqrt{3}}{3}, \infty)$

Step 4

Substitute any number from the interval $(- \infty, - \frac{\sqrt{3}}{3})$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

Replace the variable $x$ with $- 3$ in the expression.

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

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

Raise $- 3$ to the power of $2$ .

$f'' (- 3) = 12 \cdot 9 - 4$

Step 4.2.1.2

Multiply $12$ by $9$ .

$f'' (- 3) = 108 - 4$

Step 4.2.2

Subtract $4$ from $108$ .

$f'' (- 3) = 104$

Step 4.2.3

The final answer is $104$ .

$104$

Step 4.3

The graph is concave up on the interval $(- \infty, - \frac{\sqrt{3}}{3})$ because $f'' (- 3)$ is positive.

Concave up on $(- \infty, - \frac{\sqrt{3}}{3})$ since $f'' (x)$ is positive

Step 5

Substitute any number from the interval $(- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

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

$f'' (0) = 12 {(0)}^{2} - 4$

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

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

$f'' (0) = 12 \cdot 0 - 4$

Step 5.2.1.2

Multiply $12$ by $0$ .

$f'' (0) = 0 - 4$

Step 5.2.2

Subtract $4$ from $0$ .

$f'' (0) = - 4$

Step 5.2.3

The final answer is $- 4$ .

$- 4$

Step 5.3

The graph is concave down on the interval $(- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$ because $f'' (0)$ is negative.

Concave down on $(- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(\frac{\sqrt{3}}{3}, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $3$ in the expression.

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

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 $3$ to the power of $2$ .

$f'' (3) = 12 \cdot 9 - 4$

Step 6.2.1.2

Multiply $12$ by $9$ .

$f'' (3) = 108 - 4$

Step 6.2.2

Subtract $4$ from $108$ .

$f'' (3) = 104$

Step 6.2.3

The final answer is $104$ .

$104$

Step 6.3

The graph is concave up on the interval $(\frac{\sqrt{3}}{3}, \infty)$ because $f'' (3)$ is positive.

Concave up on $(\frac{\sqrt{3}}{3}, \infty)$ since $f'' (x)$ is positive

Step 7

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave up on $(- \infty, - \frac{\sqrt{3}}{3})$ since $f'' (x)$ is positive

Concave down on $(- \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$ since $f'' (x)$ is negative

Concave up on $(\frac{\sqrt{3}}{3}, \infty)$ since $f'' (x)$ is positive

Step 8