Calculus Examples

${(x^{2} - 1)}^{3}$

Step 1

Write ${(x^{2} - 1)}^{3}$ as a function.

$f (x) = {(x^{2} - 1)}^{3}$

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

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = x^{2} - 1$ .

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Step 2.1.1.1.1

To apply the Chain Rule, set $u$ as $x^{2} - 1$ .

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

Step 2.1.1.1.2

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

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

Step 2.1.1.1.3

Replace all occurrences of $u$ with $x^{2} - 1$ .

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

Step 2.1.1.2

Differentiate.

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Step 2.1.1.2.1

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

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

Step 2.1.1.2.2

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

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

Step 2.1.1.2.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$3 {(x^{2} - 1)}^{2} (2 x + 0)$

Step 2.1.1.2.4

Simplify the expression.

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Step 2.1.1.2.4.1

Add $2 x$ and $0$ .

$3 {(x^{2} - 1)}^{2} (2 x)$

Step 2.1.1.2.4.2

Multiply $2$ by $3$ .

$6 {(x^{2} - 1)}^{2} x$

Step 2.1.1.2.4.3

Reorder the factors of $6 {(x^{2} - 1)}^{2} x$ .

$f' (x) = 6 x {(x^{2} - 1)}^{2}$

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 $6 x {(x^{2} - 1)}^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x {(x^{2} - 1)}^{2}]$ .

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

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 $f (x) = x$ and $g (x) = {(x^{2} - 1)}^{2}$ .

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

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 $f (x) = x^{2}$ and $g (x) = x^{2} - 1$ .

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Step 2.1.2.3.1

To apply the Chain Rule, set $u$ as $x^{2} - 1$ .

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

Step 2.1.2.3.2

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

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

Step 2.1.2.3.3

Replace all occurrences of $u$ with $x^{2} - 1$ .

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

Step 2.1.2.4

Differentiate.

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Step 2.1.2.4.1

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

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

Step 2.1.2.4.2

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

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

Step 2.1.2.4.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

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

Step 2.1.2.4.4

Simplify the expression.

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Step 2.1.2.4.4.1

Add $2 x$ and $0$ .

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

Step 2.1.2.4.4.2

Multiply $2$ by $2$ .

$6 (x (4 (x^{2} - 1) x) + {(x^{2} - 1)}^{2} \frac{d}{d x} [x])$

Step 2.1.2.5

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

$6 (x^{1} x (4 (x^{2} - 1)) + {(x^{2} - 1)}^{2} \frac{d}{d x} [x])$

Step 2.1.2.6

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

$6 (x^{1} x^{1} (4 (x^{2} - 1)) + {(x^{2} - 1)}^{2} \frac{d}{d x} [x])$

Step 2.1.2.7

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

$6 (x^{1 + 1} (4 (x^{2} - 1)) + {(x^{2} - 1)}^{2} \frac{d}{d x} [x])$

Step 2.1.2.8

Add $1$ and $1$ .

$6 (x^{2} (4 (x^{2} - 1)) + {(x^{2} - 1)}^{2} \frac{d}{d x} [x])$

Step 2.1.2.9

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

$6 (x^{2} (4 (x^{2} - 1)) + {(x^{2} - 1)}^{2} \cdot 1)$

Step 2.1.2.10

Multiply ${(x^{2} - 1)}^{2}$ by $1$ .

$6 (x^{2} (4 (x^{2} - 1)) + {(x^{2} - 1)}^{2})$

Step 2.1.2.11

Simplify.

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Step 2.1.2.11.1

Apply the distributive property.

$6 (x^{2} (4 x^{2} + 4 \cdot - 1) + {(x^{2} - 1)}^{2})$

Step 2.1.2.11.2

Apply the distributive property.

$6 (x^{2} (4 x^{2}) + x^{2} (4 \cdot - 1) + {(x^{2} - 1)}^{2})$

Step 2.1.2.11.3

Apply the distributive property.

$6 (x^{2} (4 x^{2})) + 6 (x^{2} (4 \cdot - 1)) + 6 {(x^{2} - 1)}^{2}$

Step 2.1.2.11.4

Combine terms.

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Step 2.1.2.11.4.1

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

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Step 2.1.2.11.4.1.1

Move $x^{2}$ .

$6 (x^{2} x^{2} \cdot 4) + 6 (x^{2} (4 \cdot - 1)) + 6 {(x^{2} - 1)}^{2}$

Step 2.1.2.11.4.1.2

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

$6 (x^{2 + 2} \cdot 4) + 6 (x^{2} (4 \cdot - 1)) + 6 {(x^{2} - 1)}^{2}$

Step 2.1.2.11.4.1.3

Add $2$ and $2$ .

$6 (x^{4} \cdot 4) + 6 (x^{2} (4 \cdot - 1)) + 6 {(x^{2} - 1)}^{2}$

Step 2.1.2.11.4.2

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

$6 (4 \cdot x^{4}) + 6 (x^{2} (4 \cdot - 1)) + 6 {(x^{2} - 1)}^{2}$

Step 2.1.2.11.4.3

Multiply $4$ by $6$ .

$24 x^{4} + 6 (x^{2} (4 \cdot - 1)) + 6 {(x^{2} - 1)}^{2}$

Step 2.1.2.11.4.4

Multiply $4$ by $- 1$ .

$24 x^{4} + 6 (x^{2} \cdot - 4) + 6 {(x^{2} - 1)}^{2}$

Step 2.1.2.11.4.5

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

$24 x^{4} + 6 (- 4 \cdot x^{2}) + 6 {(x^{2} - 1)}^{2}$

Step 2.1.2.11.4.6

Multiply $- 4$ by $6$ .

$24 x^{4} - 24 x^{2} + 6 {(x^{2} - 1)}^{2}$

Step 2.1.2.11.5

Simplify each term.

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Step 2.1.2.11.5.1

Rewrite ${(x^{2} - 1)}^{2}$ as $(x^{2} - 1) (x^{2} - 1)$ .

$24 x^{4} - 24 x^{2} + 6 ((x^{2} - 1) (x^{2} - 1))$

Step 2.1.2.11.5.2

Expand $(x^{2} - 1) (x^{2} - 1)$ using the FOIL Method.

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Step 2.1.2.11.5.2.1

Apply the distributive property.

$24 x^{4} - 24 x^{2} + 6 (x^{2} (x^{2} - 1) - 1 (x^{2} - 1))$

Step 2.1.2.11.5.2.2

Apply the distributive property.

$24 x^{4} - 24 x^{2} + 6 (x^{2} x^{2} + x^{2} \cdot - 1 - 1 (x^{2} - 1))$

Step 2.1.2.11.5.2.3

Apply the distributive property.

$24 x^{4} - 24 x^{2} + 6 (x^{2} x^{2} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1)$

Step 2.1.2.11.5.3

Simplify and combine like terms.

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Step 2.1.2.11.5.3.1

Simplify each term.

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Step 2.1.2.11.5.3.1.1

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

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Step 2.1.2.11.5.3.1.1.1

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

$24 x^{4} - 24 x^{2} + 6 (x^{2 + 2} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1)$

Step 2.1.2.11.5.3.1.1.2

Add $2$ and $2$ .

$24 x^{4} - 24 x^{2} + 6 (x^{4} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1)$

Step 2.1.2.11.5.3.1.2

Move $- 1$ to the left of $x^{2}$ .

$24 x^{4} - 24 x^{2} + 6 (x^{4} - 1 \cdot x^{2} - 1 x^{2} - 1 \cdot - 1)$

Step 2.1.2.11.5.3.1.3

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

$24 x^{4} - 24 x^{2} + 6 (x^{4} - x^{2} - 1 x^{2} - 1 \cdot - 1)$

Step 2.1.2.11.5.3.1.4

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

$24 x^{4} - 24 x^{2} + 6 (x^{4} - x^{2} - x^{2} - 1 \cdot - 1)$

Step 2.1.2.11.5.3.1.5

Multiply $- 1$ by $- 1$ .

$24 x^{4} - 24 x^{2} + 6 (x^{4} - x^{2} - x^{2} + 1)$

Step 2.1.2.11.5.3.2

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

$24 x^{4} - 24 x^{2} + 6 (x^{4} - 2 x^{2} + 1)$

Step 2.1.2.11.5.4

Apply the distributive property.

$24 x^{4} - 24 x^{2} + 6 x^{4} + 6 (- 2 x^{2}) + 6 \cdot 1$

Step 2.1.2.11.5.5

Simplify.

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Step 2.1.2.11.5.5.1

Multiply $- 2$ by $6$ .

$24 x^{4} - 24 x^{2} + 6 x^{4} - 12 x^{2} + 6 \cdot 1$

Step 2.1.2.11.5.5.2

Multiply $6$ by $1$ .

$24 x^{4} - 24 x^{2} + 6 x^{4} - 12 x^{2} + 6$

Step 2.1.2.11.6

Add $24 x^{4}$ and $6 x^{4}$ .

$30 x^{4} - 24 x^{2} - 12 x^{2} + 6$

Step 2.1.2.11.7

Subtract $12 x^{2}$ from $- 24 x^{2}$ .

$f'' (x) = 30 x^{4} - 36 x^{2} + 6$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $30 x^{4} - 36 x^{2} + 6$ .

$30 x^{4} - 36 x^{2} + 6$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $30 x^{4} - 36 x^{2} + 6 = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$30 x^{4} - 36 x^{2} + 6 = 0$

Step 2.2.2

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$30 u^{2} - 36 u + 6 = 0$

$u = x^{2}$

Step 2.2.3

Factor the left side of the equation.

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Step 2.2.3.1

Factor $6$ out of $30 u^{2} - 36 u + 6$ .

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Step 2.2.3.1.1

Factor $6$ out of $30 u^{2}$ .

$6 (5 u^{2}) - 36 u + 6 = 0$

Step 2.2.3.1.2

Factor $6$ out of $- 36 u$ .

$6 (5 u^{2}) + 6 (- 6 u) + 6 = 0$

Step 2.2.3.1.3

Factor $6$ out of $6$ .

$6 (5 u^{2}) + 6 (- 6 u) + 6 (1) = 0$

Step 2.2.3.1.4

Factor $6$ out of $6 (5 u^{2}) + 6 (- 6 u)$ .

$6 (5 u^{2} - 6 u) + 6 (1) = 0$

Step 2.2.3.1.5

Factor $6$ out of $6 (5 u^{2} - 6 u) + 6 (1)$ .

$6 (5 u^{2} - 6 u + 1) = 0$

Step 2.2.3.2

Factor.

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Step 2.2.3.2.1

Factor by grouping.

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Step 2.2.3.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot 1 = 5$ and whose sum is $b = - 6$ .

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Step 2.2.3.2.1.1.1

Factor $- 6$ out of $- 6 u$ .

$6 (5 u^{2} - 6 u + 1) = 0$

Step 2.2.3.2.1.1.2

Rewrite $- 6$ as $- 1$ plus $- 5$

$6 (5 u^{2} + (- 1 - 5) u + 1) = 0$

Step 2.2.3.2.1.1.3

Apply the distributive property.

$6 (5 u^{2} - 1 u - 5 u + 1) = 0$

Step 2.2.3.2.1.2

Factor out the greatest common factor from each group.

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Step 2.2.3.2.1.2.1

Group the first two terms and the last two terms.

$6 ((5 u^{2} - 1 u) - 5 u + 1) = 0$

Step 2.2.3.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$6 (u (5 u - 1) - (5 u - 1)) = 0$

Step 2.2.3.2.1.3

Factor the polynomial by factoring out the greatest common factor, $5 u - 1$ .

$6 ((5 u - 1) (u - 1)) = 0$

Step 2.2.3.2.2

Remove unnecessary parentheses.

$6 (5 u - 1) (u - 1) = 0$

Step 2.2.4

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

$5 u - 1 = 0$

$u - 1 = 0$

Step 2.2.5

Set $5 u - 1$ equal to $0$ and solve for $u$ .

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Step 2.2.5.1

Set $5 u - 1$ equal to $0$ .

$5 u - 1 = 0$

Step 2.2.5.2

Solve $5 u - 1 = 0$ for $u$ .

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Step 2.2.5.2.1

Add $1$ to both sides of the equation.

$5 u = 1$

Step 2.2.5.2.2

Divide each term in $5 u = 1$ by $5$ and simplify.

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Step 2.2.5.2.2.1

Divide each term in $5 u = 1$ by $5$ .

$\frac{5 u}{5} = \frac{1}{5}$

Step 2.2.5.2.2.2

Simplify the left side.

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Step 2.2.5.2.2.2.1

Cancel the common factor of $5$ .

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Step 2.2.5.2.2.2.1.1

Cancel the common factor.

$\frac{5 u}{5} = \frac{1}{5}$

Step 2.2.5.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{1}{5}$

Step 2.2.6

Set $u - 1$ equal to $0$ and solve for $u$ .

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Step 2.2.6.1

Set $u - 1$ equal to $0$ .

$u - 1 = 0$

Step 2.2.6.2

Add $1$ to both sides of the equation.

$u = 1$

Step 2.2.7

The final solution is all the values that make $6 (5 u - 1) (u - 1) = 0$ true.

$u = \frac{1}{5}, 1$

Step 2.2.8

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = \frac{1}{5}$

${(x^{2})}^{1} = 1$

Step 2.2.9

Solve the first equation for $x$ .

$x^{2} = \frac{1}{5}$

Step 2.2.10

Solve the equation for $x$ .

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Step 2.2.10.1

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

$x = \pm \sqrt{\frac{1}{5}}$

Step 2.2.10.2

Simplify $\pm \sqrt{\frac{1}{5}}$ .

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Step 2.2.10.2.1

Rewrite $\sqrt{\frac{1}{5}}$ as $\frac{\sqrt{1}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{5}}$

Step 2.2.10.2.2

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{5}}$

Step 2.2.10.2.3

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 2.2.10.2.4

Combine and simplify the denominator.

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Step 2.2.10.2.4.1

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 2.2.10.2.4.2

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{\sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 2.2.10.2.4.3

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{\sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 2.2.10.2.4.4

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

$x = \pm \frac{\sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 2.2.10.2.4.5

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{5}}{{\sqrt{5}}^{2}}$

Step 2.2.10.2.4.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.2.10.2.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 2.2.10.2.4.6.2

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

$x = \pm \frac{\sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 2.2.10.2.4.6.3

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

$x = \pm \frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 2.2.10.2.4.6.4

Cancel the common factor of $2$ .

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Step 2.2.10.2.4.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{5}}{5^{\frac{2}{2}}}$

Step 2.2.10.2.4.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{5}}{5^{1}}$

Step 2.2.10.2.4.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{5}}{5}$

Step 2.2.10.3

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

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Step 2.2.10.3.1

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

$x = \frac{\sqrt{5}}{5}$

Step 2.2.10.3.2

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

$x = - \frac{\sqrt{5}}{5}$

Step 2.2.10.3.3

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

$x = \frac{\sqrt{5}}{5}, - \frac{\sqrt{5}}{5}$

Step 2.2.11

Solve the second equation for $x$ .

${(x^{2})}^{1} = 1$

Step 2.2.12

Solve the equation for $x$ .

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Step 2.2.12.1

Remove parentheses.

$x^{2} = 1$

Step 2.2.12.2

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

$x = \pm \sqrt{1}$

Step 2.2.12.3

Any root of $1$ is $1$ .

$x = \pm 1$

Step 2.2.12.4

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

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Step 2.2.12.4.1

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

$x = 1$

Step 2.2.12.4.2

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

$x = - 1$

Step 2.2.12.4.3

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

$x = 1, - 1$

Step 2.2.13

The solution to $30 x^{4} - 36 x^{2} + 6 = 0$ is $x = \frac{\sqrt{5}}{5}, - \frac{\sqrt{5}}{5}, 1, - 1$ .

$x = \frac{\sqrt{5}}{5}, - \frac{\sqrt{5}}{5}, 1, - 1$

Step 3

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 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, - 1) \cup (- 1, - \frac{\sqrt{5}}{5}) \cup (- \frac{\sqrt{5}}{5}, \frac{\sqrt{5}}{5}) \cup (\frac{\sqrt{5}}{5}, 1) \cup (1, \infty)$

Step 5

Substitute any number from the interval $(- \infty, - 1)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $- 4$ in the expression.

$f'' (- 4) = 30 {(- 4)}^{4} - 36 {(- 4)}^{2} + 6$

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

$f'' (- 4) = 30 \cdot 256 - 36 {(- 4)}^{2} + 6$

Step 5.2.1.2

Multiply $30$ by $256$ .

$f'' (- 4) = 7680 - 36 {(- 4)}^{2} + 6$

Step 5.2.1.3

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

$f'' (- 4) = 7680 - 36 \cdot 16 + 6$

Step 5.2.1.4

Multiply $- 36$ by $16$ .

$f'' (- 4) = 7680 - 576 + 6$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $576$ from $7680$ .

$f'' (- 4) = 7104 + 6$

Step 5.2.2.2

Add $7104$ and $6$ .

$f'' (- 4) = 7110$

Step 5.2.3

The final answer is $7110$ .

$7110$

Step 5.3

The graph is concave up on the interval $(- \infty, - 1)$ because $f'' (- 4)$ is positive.

Concave up on $(- \infty, - 1)$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(- 1, - \frac{\sqrt{5}}{5})$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $- 0.72$ in the expression.

$f'' (- 0.72) = 30 {(- 0.72)}^{4} - 36 {(- 0.72)}^{2} + 6$

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 $- 0.72$ to the power of $4$ .

$f'' (- 0.72) = 30 \cdot 0.26873856 - 36 {(- 0.72)}^{2} + 6$

Step 6.2.1.2

Multiply $30$ by $0.26873856$ .

$f'' (- 0.72) = 8.0621568 - 36 {(- 0.72)}^{2} + 6$

Step 6.2.1.3

Raise $- 0.72$ to the power of $2$ .

$f'' (- 0.72) = 8.0621568 - 36 \cdot 0.5184 + 6$

Step 6.2.1.4

Multiply $- 36$ by $0.5184$ .

$f'' (- 0.72) = 8.0621568 - 18.6624 + 6$

Step 6.2.2

Simplify by adding and subtracting.

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Step 6.2.2.1

Subtract $18.6624$ from $8.0621568$ .

$f'' (- 0.72) = - 10.6002432 + 6$

Step 6.2.2.2

Add $- 10.6002432$ and $6$ .

$f'' (- 0.72) = - 4.6002432$

Step 6.2.3

The final answer is $- 4.6002432$ .

$- 4.6002432$

Step 6.3

The graph is concave down on the interval $(- 1, - \frac{\sqrt{5}}{5})$ because $f'' (- 0.72)$ is negative.

Concave down on $(- 1, - \frac{\sqrt{5}}{5})$ since $f'' (x)$ is negative

Step 7

Substitute any number from the interval $(- \frac{\sqrt{5}}{5}, \frac{\sqrt{5}}{5})$ into the second derivative and evaluate to determine the concavity.

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Step 7.1

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

$f'' (0) = 30 {(0)}^{4} - 36 {(0)}^{2} + 6$

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 $0$ .

$f'' (0) = 30 \cdot 0 - 36 {(0)}^{2} + 6$

Step 7.2.1.2

Multiply $30$ by $0$ .

$f'' (0) = 0 - 36 {(0)}^{2} + 6$

Step 7.2.1.3

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

$f'' (0) = 0 - 36 \cdot 0 + 6$

Step 7.2.1.4

Multiply $- 36$ by $0$ .

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

Step 7.2.2

Simplify by adding numbers.

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Step 7.2.2.1

Add $0$ and $0$ .

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

Step 7.2.2.2

Add $0$ and $6$ .

$f'' (0) = 6$

Step 7.2.3

The final answer is $6$ .

$6$

Step 7.3

The graph is concave up on the interval $(- \frac{\sqrt{5}}{5}, \frac{\sqrt{5}}{5})$ because $f'' (0)$ is positive.

Concave up on $(- \frac{\sqrt{5}}{5}, \frac{\sqrt{5}}{5})$ since $f'' (x)$ is positive

Step 8

Substitute any number from the interval $(\frac{\sqrt{5}}{5}, 1)$ into the second derivative and evaluate to determine the concavity.

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Step 8.1

Replace the variable $x$ with $0.72$ in the expression.

$f'' (0.72) = 30 {(0.72)}^{4} - 36 {(0.72)}^{2} + 6$

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 $0.72$ to the power of $4$ .

$f'' (0.72) = 30 \cdot 0.26873856 - 36 {(0.72)}^{2} + 6$

Step 8.2.1.2

Multiply $30$ by $0.26873856$ .

$f'' (0.72) = 8.0621568 - 36 {(0.72)}^{2} + 6$

Step 8.2.1.3

Raise $0.72$ to the power of $2$ .

$f'' (0.72) = 8.0621568 - 36 \cdot 0.5184 + 6$

Step 8.2.1.4

Multiply $- 36$ by $0.5184$ .

$f'' (0.72) = 8.0621568 - 18.6624 + 6$

Step 8.2.2

Simplify by adding and subtracting.

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Step 8.2.2.1

Subtract $18.6624$ from $8.0621568$ .

$f'' (0.72) = - 10.6002432 + 6$

Step 8.2.2.2

Add $- 10.6002432$ and $6$ .

$f'' (0.72) = - 4.6002432$

Step 8.2.3

The final answer is $- 4.6002432$ .

$- 4.6002432$

Step 8.3

The graph is concave down on the interval $(\frac{\sqrt{5}}{5}, 1)$ because $f'' (0.72)$ is negative.

Concave down on $(\frac{\sqrt{5}}{5}, 1)$ since $f'' (x)$ is negative

Step 9

Substitute any number from the interval $(1, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 9.1

Replace the variable $x$ with $4$ in the expression.

$f'' (4) = 30 {(4)}^{4} - 36 {(4)}^{2} + 6$

Step 9.2

Simplify the result.

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Step 9.2.1

Simplify each term.

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Step 9.2.1.1

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

$f'' (4) = 30 \cdot 256 - 36 {(4)}^{2} + 6$

Step 9.2.1.2

Multiply $30$ by $256$ .

$f'' (4) = 7680 - 36 {(4)}^{2} + 6$

Step 9.2.1.3

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

$f'' (4) = 7680 - 36 \cdot 16 + 6$

Step 9.2.1.4

Multiply $- 36$ by $16$ .

$f'' (4) = 7680 - 576 + 6$

Step 9.2.2

Simplify by adding and subtracting.

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Step 9.2.2.1

Subtract $576$ from $7680$ .

$f'' (4) = 7104 + 6$

Step 9.2.2.2

Add $7104$ and $6$ .

$f'' (4) = 7110$

Step 9.2.3

The final answer is $7110$ .

$7110$

Step 9.3

The graph is concave up on the interval $(1, \infty)$ because $f'' (4)$ is positive.

Concave up on $(1, \infty)$ since $f'' (x)$ is positive

Step 10

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

Concave up on $(- \infty, - 1)$ since $f'' (x)$ is positive

Concave down on $(- 1, - \frac{\sqrt{5}}{5})$ since $f'' (x)$ is negative

Concave up on $(- \frac{\sqrt{5}}{5}, \frac{\sqrt{5}}{5})$ since $f'' (x)$ is positive

Concave down on $(\frac{\sqrt{5}}{5}, 1)$ since $f'' (x)$ is negative

Concave up on $(1, \infty)$ since $f'' (x)$ is positive

Step 11