Calculus Examples

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

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 using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2} + 12$ and $g (x) = 4 - x^{2}$ .

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

Step 1.1.1.2

Differentiate.

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Step 1.1.1.2.1

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

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

Step 1.1.1.2.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$f' (x) = (x^{2} + 12) (0 + \frac{d}{d x} (- x^{2})) + (4 - x^{2}) \frac{d}{d x} (x^{2} + 12)$

Step 1.1.1.2.3

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

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

Step 1.1.1.2.4

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

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

Step 1.1.1.2.5

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

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

Step 1.1.1.2.6

Multiply $2$ by $- 1$ .

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

Step 1.1.1.2.7

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

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

Step 1.1.1.2.8

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

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

Step 1.1.1.2.9

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

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

Step 1.1.1.2.10

Simplify the expression.

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Step 1.1.1.2.10.1

Add $2 x$ and $0$ .

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

Step 1.1.1.2.10.2

Move $2$ to the left of $4 - x^{2}$ .

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

Step 1.1.1.3

Simplify.

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Step 1.1.1.3.1

Apply the distributive property.

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

Step 1.1.1.3.2

Apply the distributive property.

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

Step 1.1.1.3.3

Apply the distributive property.

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

Step 1.1.1.3.4

Combine terms.

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Step 1.1.1.3.4.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.1.1.3.4.1.1

Move $x$ .

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

Step 1.1.1.3.4.1.2

Multiply $x$ by $x^{2}$ .

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Step 1.1.1.3.4.1.2.1

Raise $x$ to the power of $1$ .

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

Step 1.1.1.3.4.1.2.2

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

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

Step 1.1.1.3.4.1.3

Add $1$ and $2$ .

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

Step 1.1.1.3.4.2

Move $- 2$ to the left of $x^{3}$ .

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

Step 1.1.1.3.4.3

Multiply $- 2$ by $12$ .

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

Step 1.1.1.3.4.4

Multiply $2$ by $4$ .

$f' (x) = - 2 x^{3} - 24 x + 8 x + 2 (- x^{2}) x$

Step 1.1.1.3.4.5

Multiply $- 1$ by $2$ .

$f' (x) = - 2 x^{3} - 24 x + 8 x - 2 x^{2} x$

Step 1.1.1.3.4.6

Raise $x$ to the power of $1$ .

$f' (x) = - 2 x^{3} - 24 x + 8 x - 2 (x \cdot x^{2})$

Step 1.1.1.3.4.7

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

$f' (x) = - 2 x^{3} - 24 x + 8 x - 2 x^{1 + 2}$

Step 1.1.1.3.4.8

Add $1$ and $2$ .

$f' (x) = - 2 x^{3} - 24 x + 8 x - 2 x^{3}$

Step 1.1.1.3.4.9

Add $- 24 x$ and $8 x$ .

$f' (x) = - 2 x^{3} - 16 x - 2 x^{3}$

Step 1.1.1.3.4.10

Subtract $2 x^{3}$ from $- 2 x^{3}$ .

$f' (x) = - 4 x^{3} - 16 x$

Step 1.1.2

Find the second derivative.

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Step 1.1.2.1

By the Sum Rule, the derivative of $- 4 x^{3} - 16 x$ with respect to $x$ is $\frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [- 16 x]$ .

$f'' (x) = \frac{d}{d x} (- 4 x^{3}) + \frac{d}{d x} (- 16 x)$

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

$f'' (x) = - 4 \frac{d}{d x} x^{3} + \frac{d}{d x} (- 16 x)$

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 $n = 3$ .

$f'' (x) = - 4 (3 x^{2}) + \frac{d}{d x} (- 16 x)$

Step 1.1.2.2.3

Multiply $3$ by $- 4$ .

$f'' (x) = - 12 x^{2} + \frac{d}{d x} (- 16 x)$

Step 1.1.2.3

Evaluate $\frac{d}{d x} [- 16 x]$ .

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Step 1.1.2.3.1

Since $- 16$ is constant with respect to $x$ , the derivative of $- 16 x$ with respect to $x$ is $- 16 \frac{d}{d x} [x]$ .

$f'' (x) = - 12 x^{2} - 16 \frac{d}{d x} x$

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 $n = 1$ .

$f'' (x) = - 12 x^{2} - 16 \cdot 1$

Step 1.1.2.3.3

Multiply $- 16$ by $1$ .

$f'' (x) = - 12 x^{2} - 16$

Step 1.1.3

The second derivative of $f (x)$ with respect to $x$ is $- 12 x^{2} - 16$ .

$- 12 x^{2} - 16$

Step 1.2

Set the second derivative equal to $0$ then solve the equation $- 12 x^{2} - 16 = 0$ .

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Step 1.2.1

Set the second derivative equal to $0$ .

$- 12 x^{2} - 16 = 0$

Step 1.2.2

Add $16$ to both sides of the equation.

$- 12 x^{2} = 16$

Step 1.2.3

Divide each term in $- 12 x^{2} = 16$ by $- 12$ and simplify.

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Step 1.2.3.1

Divide each term in $- 12 x^{2} = 16$ by $- 12$ .

$\frac{- 12 x^{2}}{- 12} = \frac{16}{- 12}$

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.

$\frac{- 12 x^{2}}{- 12} = \frac{16}{- 12}$

Step 1.2.3.2.1.2

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

$x^{2} = \frac{16}{- 12}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Cancel the common factor of $16$ and $- 12$ .

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Step 1.2.3.3.1.1

Factor $4$ out of $16$ .

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

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 $- 12$ .

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

Step 1.2.3.3.1.2.2

Cancel the common factor.

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

Step 1.2.3.3.1.2.3

Rewrite the expression.

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

Step 1.2.3.3.2

Move the negative in front of the fraction.

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

Step 1.2.4

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

$x = \pm \sqrt{- \frac{4}{3}}$

Step 1.2.5

Simplify $\pm \sqrt{- \frac{4}{3}}$ .

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Step 1.2.5.1

Rewrite $- \frac{4}{3}$ as $i^{2} \frac{2^{2}}{3}$ .

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Step 1.2.5.1.1

Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{4}{3}}$

Step 1.2.5.1.2

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

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

Step 1.2.5.2

Pull terms out from under the radical.

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

Step 1.2.5.3

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

$x = \pm i \sqrt{\frac{4}{3}}$

Step 1.2.5.4

Rewrite $\sqrt{\frac{4}{3}}$ as $\frac{\sqrt{4}}{\sqrt{3}}$ .

$x = \pm i \frac{\sqrt{4}}{\sqrt{3}}$

Step 1.2.5.5

Simplify the numerator.

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Step 1.2.5.5.1

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

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

Step 1.2.5.5.2

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

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

Step 1.2.5.6

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm i (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 1.2.5.7

Combine and simplify the denominator.

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Step 1.2.5.7.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 1.2.5.7.2

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

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

Step 1.2.5.7.3

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

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

Step 1.2.5.7.4

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

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

Step 1.2.5.7.5

Add $1$ and $1$ .

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

Step 1.2.5.7.6

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

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Step 1.2.5.7.6.1

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

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

Step 1.2.5.7.6.2

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

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

Step 1.2.5.7.6.3

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

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

Step 1.2.5.7.6.4

Cancel the common factor of $2$ .

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Step 1.2.5.7.6.4.1

Cancel the common factor.

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

Step 1.2.5.7.6.4.2

Rewrite the expression.

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

Step 1.2.5.7.6.5

Evaluate the exponent.

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

Step 1.2.5.8

Combine $i$ and $\frac{2 \sqrt{3}}{3}$ .

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

Step 1.2.5.9

Move $2$ to the left of $i$ .

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

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.

$x = \frac{2 i \sqrt{3}}{3}$

Step 1.2.6.2

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

$x = - \frac{2 i \sqrt{3}}{3}$

Step 1.2.6.3

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

$x = \frac{2 i \sqrt{3}}{3}, - \frac{2 i \sqrt{3}}{3}$

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.

$(- \infty, \infty)$

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.

$f'' (0) = - 12 {(0)}^{2} - 16$

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 $0$ .

$f'' (0) = - 12 \cdot 0 - 16$

Step 4.2.1.2

Multiply $- 12$ by $0$ .

$f'' (0) = 0 - 16$

Step 4.2.2

Subtract $16$ from $0$ .

$f'' (0) = - 16$

Step 4.2.3

The final answer is $- 16$ .

$- 16$

Step 4.3

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

The graph is concave down

Step 5