Calculus Examples

$6 \sec (x - \frac{π}{2})$

Step 1

Write $6 \sec (x - \frac{π}{2})$ as a function.

$f (x) = 6 \sec (x - \frac{π}{2})$

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

Since $6$ is constant with respect to $x$ , the derivative of $6 \sec (x - \frac{π}{2})$ with respect to $x$ is $6 \frac{d}{d x} [\sec (x - \frac{π}{2})]$ .

$6 \frac{d}{d x} [\sec (x - \frac{π}{2})]$

Step 2.1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = x - \frac{π}{2}$ .

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Step 2.1.1.2.1

To apply the Chain Rule, set $u$ as $x - \frac{π}{2}$ .

$6 (\frac{d}{d u} [\sec (u)] \frac{d}{d x} [x - \frac{π}{2}])$

Step 2.1.1.2.2

The derivative of $\sec (u)$ with respect to $u$ is $\sec (u) \tan (u)$ .

$6 (\sec (u) \tan (u) \frac{d}{d x} [x - \frac{π}{2}])$

Step 2.1.1.2.3

Replace all occurrences of $u$ with $x - \frac{π}{2}$ .

$6 (\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) \frac{d}{d x} [x - \frac{π}{2}])$

Step 2.1.1.3

Differentiate.

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Step 2.1.1.3.1

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

$6 \sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (\frac{d}{d x} [x] + \frac{d}{d x} [- \frac{π}{2}])$

Step 2.1.1.3.2

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

$6 \sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + \frac{d}{d x} [- \frac{π}{2}])$

Step 2.1.1.3.3

Since $- \frac{π}{2}$ is constant with respect to $x$ , the derivative of $- \frac{π}{2}$ with respect to $x$ is $0$ .

$6 \sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) (1 + 0)$

Step 2.1.1.3.4

Simplify the expression.

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Step 2.1.1.3.4.1

Add $1$ and $0$ .

$6 \sec (x - \frac{π}{2}) \tan (x - \frac{π}{2}) \cdot 1$

Step 2.1.1.3.4.2

Multiply $6$ by $1$ .

$f' (x) = 6 \sec (x - \frac{π}{2}) \tan (x - \frac{π}{2})$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 \sec (x - \frac{π}{2}) \tan (x - \frac{π}{2})$ with respect to $x$ is $6 \frac{d}{d x} [\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$ .

$6 \frac{d}{d x} [\sec (x - \frac{π}{2}) \tan (x - \frac{π}{2})]$

Step 2.1.2.2

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 $f (x) = \sec (x - \frac{π}{2})$ and $g (x) = \tan (x - \frac{π}{2})$ .

$6 (\sec (x - \frac{π}{2}) \frac{d}{d x} [\tan (x - \frac{π}{2})] + \tan (x - \frac{π}{2}) \frac{d}{d x} [\sec (x - \frac{π}{2})])$

Step 2.1.2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = x - \frac{π}{2}$ .

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Step 2.1.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x - \frac{π}{2}$ .

$6 (\sec (x - \frac{π}{2}) (\frac{d}{d u_{1}} [\tan (u_{1})] \frac{d}{d x} [x - \frac{π}{2}]) + \tan (x - \frac{π}{2}) \frac{d}{d x} [\sec (x - \frac{π}{2})])$

Step 2.1.2.3.2

The derivative of $\tan (u_{1})$ with respect to $u_{1}$ is $\sec^{2} (u_{1})$ .

$6 (\sec (x - \frac{π}{2}) (\sec^{2} (u_{1}) \frac{d}{d x} [x - \frac{π}{2}]) + \tan (x - \frac{π}{2}) \frac{d}{d x} [\sec (x - \frac{π}{2})])$

Step 2.1.2.3.3

Replace all occurrences of $u_{1}$ with $x - \frac{π}{2}$ .

$6 (\sec (x - \frac{π}{2}) (\sec^{2} (x - \frac{π}{2}) \frac{d}{d x} [x - \frac{π}{2}]) + \tan (x - \frac{π}{2}) \frac{d}{d x} [\sec (x - \frac{π}{2})])$

Step 2.1.2.4

Raise $\sec (x - \frac{π}{2})$ to the power of $1$ .

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

Step 2.1.2.5

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

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

Step 2.1.2.6

Differentiate.

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Step 2.1.2.6.1

Add $2$ and $1$ .

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

Step 2.1.2.6.2

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

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

Step 2.1.2.6.3

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

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

Step 2.1.2.6.4

Since $- \frac{π}{2}$ is constant with respect to $x$ , the derivative of $- \frac{π}{2}$ with respect to $x$ is $0$ .

$6 (\sec^{3} (x - \frac{π}{2}) (1 + 0) + \tan (x - \frac{π}{2}) \frac{d}{d x} [\sec (x - \frac{π}{2})])$

Step 2.1.2.6.5

Simplify the expression.

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Step 2.1.2.6.5.1

Add $1$ and $0$ .

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

Step 2.1.2.6.5.2

Multiply $\sec^{3} (x - \frac{π}{2})$ by $1$ .

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

Step 2.1.2.7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sec (x)$ and $g (x) = x - \frac{π}{2}$ .

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Step 2.1.2.7.1

To apply the Chain Rule, set $u_{2}$ as $x - \frac{π}{2}$ .

$6 (\sec^{3} (x - \frac{π}{2}) + \tan (x - \frac{π}{2}) (\frac{d}{d u_{2}} [\sec (u_{2})] \frac{d}{d x} [x - \frac{π}{2}]))$

Step 2.1.2.7.2

The derivative of $\sec (u_{2})$ with respect to $u_{2}$ is $\sec (u_{2}) \tan (u_{2})$ .

$6 (\sec^{3} (x - \frac{π}{2}) + \tan (x - \frac{π}{2}) (\sec (u_{2}) \tan (u_{2}) \frac{d}{d x} [x - \frac{π}{2}]))$

Step 2.1.2.7.3

Replace all occurrences of $u_{2}$ with $x - \frac{π}{2}$ .

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

Step 2.1.2.8

Raise $\tan (x - \frac{π}{2})$ to the power of $1$ .

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

Step 2.1.2.9

Raise $\tan (x - \frac{π}{2})$ to the power of $1$ .

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

Step 2.1.2.10

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

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

Step 2.1.2.11

Add $1$ and $1$ .

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

Step 2.1.2.12

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

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

Step 2.1.2.13

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

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

Step 2.1.2.14

Since $- \frac{π}{2}$ is constant with respect to $x$ , the derivative of $- \frac{π}{2}$ with respect to $x$ is $0$ .

$6 (\sec^{3} (x - \frac{π}{2}) + \tan^{2} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) (1 + 0))$

Step 2.1.2.15

Simplify the expression.

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Step 2.1.2.15.1

Add $1$ and $0$ .

$6 (\sec^{3} (x - \frac{π}{2}) + \tan^{2} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) \cdot 1)$

Step 2.1.2.15.2

Multiply $\tan^{2} (x - \frac{π}{2})$ by $1$ .

$6 (\sec^{3} (x - \frac{π}{2}) + \tan^{2} (x - \frac{π}{2}) \sec (x - \frac{π}{2}))$

Step 2.1.2.16

Simplify.

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Step 2.1.2.16.1

Apply the distributive property.

$6 \sec^{3} (x - \frac{π}{2}) + 6 (\tan^{2} (x - \frac{π}{2}) \sec (x - \frac{π}{2}))$

Step 2.1.2.16.2

Reorder terms.

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

Step 2.1.3

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

$6 \tan^{2} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 6 \sec^{3} (x - \frac{π}{2})$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $6 \tan^{2} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 6 \sec^{3} (x - \frac{π}{2}) = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$6 \tan^{2} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) + 6 \sec^{3} (x - \frac{π}{2}) = 0$

Step 2.2.2

Replace the $\tan^{2} (x - \frac{π}{2})$ with $\sec^{2} (x - \frac{π}{2}) - 1$ based on the $\tan^{2} (x) + 1 = \sec^{2} (x)$ identity.

$(\sec^{2} (x - \frac{π}{2}) - 1) \sec (x - \frac{π}{2}) + 6 \sec^{3} (x - \frac{π}{2}) = 0$

Step 2.2.3

Simplify each term.

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Step 2.2.3.1

Apply the distributive property.

$\sec^{2} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) - 1 \sec (x - \frac{π}{2}) + 6 \sec^{3} (x - \frac{π}{2}) = 0$

Step 2.2.3.2

Multiply $\sec^{2} (x - \frac{π}{2})$ by $\sec (x - \frac{π}{2})$ by adding the exponents.

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Step 2.2.3.2.1

Multiply $\sec^{2} (x - \frac{π}{2})$ by $\sec (x - \frac{π}{2})$ .

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Step 2.2.3.2.1.1

Raise $\sec (x - \frac{π}{2})$ to the power of $1$ .

$\sec^{2} (x - \frac{π}{2}) \sec (x - \frac{π}{2}) - 1 \sec (x - \frac{π}{2}) + 6 \sec^{3} (x - \frac{π}{2}) = 0$

Step 2.2.3.2.1.2

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

${\sec (x - \frac{π}{2})}^{2 + 1} - 1 \sec (x - \frac{π}{2}) + 6 \sec^{3} (x - \frac{π}{2}) = 0$

Step 2.2.3.2.2

Add $2$ and $1$ .

$\sec^{3} (x - \frac{π}{2}) - 1 \sec (x - \frac{π}{2}) + 6 \sec^{3} (x - \frac{π}{2}) = 0$

Step 2.2.3.3

Rewrite $- 1 \sec (x - \frac{π}{2})$ as $- \sec (x - \frac{π}{2})$ .

$\sec^{3} (x - \frac{π}{2}) - \sec (x - \frac{π}{2}) + 6 \sec^{3} (x - \frac{π}{2}) = 0$

Step 2.2.4

Add $\sec^{3} (x - \frac{π}{2})$ and $6 \sec^{3} (x - \frac{π}{2})$ .

$7 \sec^{3} (x - \frac{π}{2}) - \sec (x - \frac{π}{2}) = 0$

Step 2.2.5

Substitute $u$ for $\sec (x - \frac{π}{2})$ .

$7 {(u)}^{3} - (u) = 0$

Step 2.2.6

Factor $u$ out of $7 u^{3} - u$ .

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Step 2.2.6.1

Factor $u$ out of $7 u^{3}$ .

$u (7 u^{2}) - u = 0$

Step 2.2.6.2

Factor $u$ out of $- u$ .

$u (7 u^{2}) + u \cdot - 1 = 0$

Step 2.2.6.3

Factor $u$ out of $u (7 u^{2}) + u \cdot - 1$ .

$u (7 u^{2} - 1) = 0$

Step 2.2.7

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$u = 0$

$7 u^{2} - 1 = 0$

Step 2.2.8

Set $u$ equal to $0$ .

$u = 0$

Step 2.2.9

Set $7 u^{2} - 1$ equal to $0$ and solve for $u$ .

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Step 2.2.9.1

Set $7 u^{2} - 1$ equal to $0$ .

$7 u^{2} - 1 = 0$

Step 2.2.9.2

Solve $7 u^{2} - 1 = 0$ for $u$ .

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Step 2.2.9.2.1

Add $1$ to both sides of the equation.

$7 u^{2} = 1$

Step 2.2.9.2.2

Divide each term in $7 u^{2} = 1$ by $7$ and simplify.

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Step 2.2.9.2.2.1

Divide each term in $7 u^{2} = 1$ by $7$ .

$\frac{7 u^{2}}{7} = \frac{1}{7}$

Step 2.2.9.2.2.2

Simplify the left side.

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Step 2.2.9.2.2.2.1

Cancel the common factor of $7$ .

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Step 2.2.9.2.2.2.1.1

Cancel the common factor.

$\frac{7 u^{2}}{7} = \frac{1}{7}$

Step 2.2.9.2.2.2.1.2

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

$u^{2} = \frac{1}{7}$

Step 2.2.9.2.3

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

$u = \pm \sqrt{\frac{1}{7}}$

Step 2.2.9.2.4

Simplify $\pm \sqrt{\frac{1}{7}}$ .

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Step 2.2.9.2.4.1

Rewrite $\sqrt{\frac{1}{7}}$ as $\frac{\sqrt{1}}{\sqrt{7}}$ .

$u = \pm \frac{\sqrt{1}}{\sqrt{7}}$

Step 2.2.9.2.4.2

Any root of $1$ is $1$ .

$u = \pm \frac{1}{\sqrt{7}}$

Step 2.2.9.2.4.3

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$u = \pm \frac{1}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 2.2.9.2.4.4

Combine and simplify the denominator.

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Step 2.2.9.2.4.4.1

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$u = \pm \frac{\sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 2.2.9.2.4.4.2

Raise $\sqrt{7}$ to the power of $1$ .

$u = \pm \frac{\sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 2.2.9.2.4.4.3

Raise $\sqrt{7}$ to the power of $1$ .

$u = \pm \frac{\sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 2.2.9.2.4.4.4

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

$u = \pm \frac{\sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 2.2.9.2.4.4.5

Add $1$ and $1$ .

$u = \pm \frac{\sqrt{7}}{{\sqrt{7}}^{2}}$

Step 2.2.9.2.4.4.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 2.2.9.2.4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$u = \pm \frac{\sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 2.2.9.2.4.4.6.2

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

$u = \pm \frac{\sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 2.2.9.2.4.4.6.3

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

$u = \pm \frac{\sqrt{7}}{7^{\frac{2}{2}}}$

Step 2.2.9.2.4.4.6.4

Cancel the common factor of $2$ .

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Step 2.2.9.2.4.4.6.4.1

Cancel the common factor.

$u = \pm \frac{\sqrt{7}}{7^{\frac{2}{2}}}$

Step 2.2.9.2.4.4.6.4.2

Rewrite the expression.

$u = \pm \frac{\sqrt{7}}{7^{1}}$

Step 2.2.9.2.4.4.6.5

Evaluate the exponent.

$u = \pm \frac{\sqrt{7}}{7}$

Step 2.2.9.2.5

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

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Step 2.2.9.2.5.1

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

$u = \frac{\sqrt{7}}{7}$

Step 2.2.9.2.5.2

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

$u = - \frac{\sqrt{7}}{7}$

Step 2.2.9.2.5.3

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

$u = \frac{\sqrt{7}}{7}, - \frac{\sqrt{7}}{7}$

Step 2.2.10

The final solution is all the values that make $u (7 u^{2} - 1) = 0$ true.

$u = 0, \frac{\sqrt{7}}{7}, - \frac{\sqrt{7}}{7}$

Step 2.2.11

Substitute $\sec (x - \frac{π}{2})$ for $u$ .

$\sec (x - \frac{π}{2}) = 0, \frac{\sqrt{7}}{7}, - \frac{\sqrt{7}}{7}$

Step 2.2.12

Set up each of the solutions to solve for $x$ .

$\sec (x - \frac{π}{2}) = 0$

$\sec (x - \frac{π}{2}) = \frac{\sqrt{7}}{7}$

$\sec (x - \frac{π}{2}) = - \frac{\sqrt{7}}{7}$

Step 2.2.13

Solve for $x$ in $\sec (x - \frac{π}{2}) = 0$ .

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Step 2.2.13.1

The range of secant is $y \leq - 1$ and $y \geq 1$ . Since $0$ does not fall in this range, there is no solution.

No solution

Step 2.2.14

Solve for $x$ in $\sec (x - \frac{π}{2}) = \frac{\sqrt{7}}{7}$ .

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Step 2.2.14.1

The range of secant is $y \leq - 1$ and $y \geq 1$ . Since $\frac{\sqrt{7}}{7}$ does not fall in this range, there is no solution.

No solution

Step 2.2.15

Solve for $x$ in $\sec (x - \frac{π}{2}) = - \frac{\sqrt{7}}{7}$ .

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Step 2.2.15.1

The range of secant is $y \leq - 1$ and $y \geq 1$ . Since $- \frac{\sqrt{7}}{7}$ does not fall in this range, there is no solution.

No solution

Step 3

Find the domain of $f (x) = 6 \sec (x - \frac{π}{2})$ .

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Step 3.1

Set the argument in $\sec (x - \frac{π}{2})$ equal to $\frac{π}{2} + π n$ to find where the expression is undefined.

$x - \frac{π}{2} = \frac{π}{2} + π n$ , for any integer $n$

Step 3.2

Move all terms not containing $x$ to the right side of the equation.

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Step 3.2.1

Add $\frac{π}{2}$ to both sides of the equation.

$x = \frac{π}{2} + π n + \frac{π}{2}$

Step 3.2.2

Combine the numerators over the common denominator.

$x = π n + \frac{π + π}{2}$

Step 3.2.3

Add $π$ and $π$ .

$x = π n + \frac{2 π}{2}$

Step 3.2.4

Cancel the common factor of $2$ .

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Step 3.2.4.1

Cancel the common factor.

$x = π n + \frac{2 π}{2}$

Step 3.2.4.2

Divide $π$ by $1$ .

$x = π n + π$

Step 3.3

The domain is all values of $x$ that make the expression defined.

Set-Builder Notation:

${x | x \neq π n + π}$ , for any integer $n$

Set-Builder Notation:

${x | x \neq π n + π}$ , for any integer $n$

Step 4

Substitute any number from the interval ${x | x \neq π n + π}$ into the second derivative and evaluate to determine the concavity.

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Step 4.1

Replace the variable $x$ with $N a N$ in the expression.

$f'' (N a N) = 6 \tan^{2} ((N a N) - \frac{π}{2}) \sec ((N a N) - \frac{π}{2}) + 6 \sec^{3} ((N a N) - \frac{π}{2})$

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

Multiply $N$ by $N$ by adding the exponents.

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Step 4.2.1.1.1

Move $N$ .

$f'' (N a N) = 6 \tan^{2} (N \cdot N a - \frac{π}{2}) \sec ((N a N) - \frac{π}{2}) + 6 \sec^{3} ((N a N) - \frac{π}{2})$

Step 4.2.1.1.2

Multiply $N$ by $N$ .

$f'' (N a N) = 6 \tan^{2} (N^{2} a - \frac{π}{2}) \sec ((N a N) - \frac{π}{2}) + 6 \sec^{3} ((N a N) - \frac{π}{2})$

Step 4.2.1.2

Multiply $N$ by $N$ by adding the exponents.

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Step 4.2.1.2.1

Move $N$ .

$f'' (N a N) = 6 \tan^{2} (N^{2} a - \frac{π}{2}) \sec (N \cdot N a - \frac{π}{2}) + 6 \sec^{3} ((N a N) - \frac{π}{2})$

Step 4.2.1.2.2

Multiply $N$ by $N$ .

$f'' (N a N) = 6 \tan^{2} (N^{2} a - \frac{π}{2}) \sec (N^{2} a - \frac{π}{2}) + 6 \sec^{3} ((N a N) - \frac{π}{2})$

Step 4.2.1.3

Multiply $N$ by $N$ by adding the exponents.

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Step 4.2.1.3.1

Move $N$ .

$f'' (N a N) = 6 \tan^{2} (N^{2} a - \frac{π}{2}) \sec (N^{2} a - \frac{π}{2}) + 6 \sec^{3} (N \cdot N a - \frac{π}{2})$

Step 4.2.1.3.2

Multiply $N$ by $N$ .

$f'' (N a N) = 6 \tan^{2} (N^{2} a - \frac{π}{2}) \sec (N^{2} a - \frac{π}{2}) + 6 \sec^{3} (N^{2} a - \frac{π}{2})$

Step 4.2.2

The final answer is $6 \tan^{2} (N^{2} a - \frac{π}{2}) \sec (N^{2} a - \frac{π}{2}) + 6 \sec^{3} (N^{2} a - \frac{π}{2})$ .

$6 \tan^{2} (N^{2} a - \frac{π}{2}) \sec (N^{2} a - \frac{π}{2}) + 6 \sec^{3} (N^{2} a - \frac{π}{2})$

Step 4.3

The graph is concave up on the interval ${x | x \neq π n + π}$ because $f'' (N a N)$ is positive.

Concave up on ${x | x \neq π n + π}$ since $f'' (x)$ is positive

Step 5