Algebra Examples

$f (x) = (x^{2} + 9) (81 - x^{2})$

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 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 $f (x) = x^{2} + 9$ and $g (x) = 81 - x^{2}$ .

$(x^{2} + 9) \frac{d}{d x} [81 - x^{2}] + (81 - x^{2}) \frac{d}{d x} [x^{2} + 9]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $81 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [81] + \frac{d}{d x} [- x^{2}]$ .

$(x^{2} + 9) (\frac{d}{d x} [81] + \frac{d}{d x} [- x^{2}]) + (81 - x^{2}) \frac{d}{d x} [x^{2} + 9]$

Step 1.1.2.2

Since $81$ is constant with respect to $x$ , the derivative of $81$ with respect to $x$ is $0$ .

$(x^{2} + 9) (0 + \frac{d}{d x} [- x^{2}]) + (81 - x^{2}) \frac{d}{d x} [x^{2} + 9]$

Step 1.1.2.3

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

$(x^{2} + 9) \frac{d}{d x} [- x^{2}] + (81 - x^{2}) \frac{d}{d x} [x^{2} + 9]$

Step 1.1.2.4

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

$(x^{2} + 9) (- \frac{d}{d x} [x^{2}]) + (81 - x^{2}) \frac{d}{d x} [x^{2} + 9]$

Step 1.1.2.5

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

$(x^{2} + 9) (- (2 x)) + (81 - x^{2}) \frac{d}{d x} [x^{2} + 9]$

Step 1.1.2.6

Multiply $2$ by $- 1$ .

$(x^{2} + 9) (- 2 x) + (81 - x^{2}) \frac{d}{d x} [x^{2} + 9]$

Step 1.1.2.7

By the Sum Rule, the derivative of $x^{2} + 9$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9]$ .

$(x^{2} + 9) (- 2 x) + (81 - x^{2}) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [9])$

Step 1.1.2.8

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

$(x^{2} + 9) (- 2 x) + (81 - x^{2}) (2 x + \frac{d}{d x} [9])$

Step 1.1.2.9

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$(x^{2} + 9) (- 2 x) + (81 - x^{2}) (2 x + 0)$

Step 1.1.2.10

Simplify the expression.

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Step 1.1.2.10.1

Add $2 x$ and $0$ .

$(x^{2} + 9) (- 2 x) + (81 - x^{2}) (2 x)$

Step 1.1.2.10.2

Move $2$ to the left of $81 - x^{2}$ .

$(x^{2} + 9) (- 2 x) + 2 \cdot (81 - x^{2}) x$

Step 1.1.3

Simplify.

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Step 1.1.3.1

Apply the distributive property.

$x^{2} (- 2 x) + 9 (- 2 x) + 2 (81 - x^{2}) x$

Step 1.1.3.2

Apply the distributive property.

$x^{2} (- 2 x) + 9 (- 2 x) + (2 \cdot 81 + 2 (- x^{2})) x$

Step 1.1.3.3

Apply the distributive property.

$x^{2} (- 2 x) + 9 (- 2 x) + 2 \cdot 81 x + 2 (- x^{2}) x$

Step 1.1.3.4

Combine terms.

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Step 1.1.3.4.1

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

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Step 1.1.3.4.1.1

Move $x$ .

$x \cdot x^{2} \cdot - 2 + 9 (- 2 x) + 2 \cdot 81 x + 2 (- x^{2}) x$

Step 1.1.3.4.1.2

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

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Step 1.1.3.4.1.2.1

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

$x^{1} x^{2} \cdot - 2 + 9 (- 2 x) + 2 \cdot 81 x + 2 (- x^{2}) x$

Step 1.1.3.4.1.2.2

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

$x^{1 + 2} \cdot - 2 + 9 (- 2 x) + 2 \cdot 81 x + 2 (- x^{2}) x$

Step 1.1.3.4.1.3

Add $1$ and $2$ .

$x^{3} \cdot - 2 + 9 (- 2 x) + 2 \cdot 81 x + 2 (- x^{2}) x$

Step 1.1.3.4.2

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

$- 2 \cdot x^{3} + 9 (- 2 x) + 2 \cdot 81 x + 2 (- x^{2}) x$

Step 1.1.3.4.3

Multiply $- 2$ by $9$ .

$- 2 x^{3} - 18 x + 2 \cdot 81 x + 2 (- x^{2}) x$

Step 1.1.3.4.4

Multiply $2$ by $81$ .

$- 2 x^{3} - 18 x + 162 x + 2 (- x^{2}) x$

Step 1.1.3.4.5

Multiply $- 1$ by $2$ .

$- 2 x^{3} - 18 x + 162 x - 2 x^{2} x$

Step 1.1.3.4.6

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

$- 2 x^{3} - 18 x + 162 x - 2 (x^{1} x^{2})$

Step 1.1.3.4.7

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

$- 2 x^{3} - 18 x + 162 x - 2 x^{1 + 2}$

Step 1.1.3.4.8

Add $1$ and $2$ .

$- 2 x^{3} - 18 x + 162 x - 2 x^{3}$

Step 1.1.3.4.9

Add $- 18 x$ and $162 x$ .

$- 2 x^{3} + 144 x - 2 x^{3}$

Step 1.1.3.4.10

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

$f' (x) = - 4 x^{3} + 144 x$

Step 1.2

Find the second derivative.

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Step 1.2.1

By the Sum Rule, the derivative of $- 4 x^{3} + 144 x$ with respect to $x$ is $\frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [144 x]$ .

$\frac{d}{d x} [- 4 x^{3}] + \frac{d}{d x} [144 x]$

Step 1.2.2

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

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Step 1.2.2.1

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

$- 4 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [144 x]$

Step 1.2.2.2

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

$- 4 (3 x^{2}) + \frac{d}{d x} [144 x]$

Step 1.2.2.3

Multiply $3$ by $- 4$ .

$- 12 x^{2} + \frac{d}{d x} [144 x]$

Step 1.2.3

Evaluate $\frac{d}{d x} [144 x]$ .

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Step 1.2.3.1

Since $144$ is constant with respect to $x$ , the derivative of $144 x$ with respect to $x$ is $144 \frac{d}{d x} [x]$ .

$- 12 x^{2} + 144 \frac{d}{d x} [x]$

Step 1.2.3.2

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

$- 12 x^{2} + 144 \cdot 1$

Step 1.2.3.3

Multiply $144$ by $1$ .

$f'' (x) = - 12 x^{2} + 144$

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $- 12 x^{2} + 144$ .

$- 12 x^{2} + 144$

Step 2

Set the second derivative equal to $0$ then solve the equation $- 12 x^{2} + 144 = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$- 12 x^{2} + 144 = 0$

Step 2.2

Subtract $144$ from both sides of the equation.

$- 12 x^{2} = - 144$

Step 2.3

Divide each term in $- 12 x^{2} = - 144$ by $- 12$ and simplify.

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Step 2.3.1

Divide each term in $- 12 x^{2} = - 144$ by $- 12$ .

$\frac{- 12 x^{2}}{- 12} = \frac{- 144}{- 12}$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $- 12$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{- 12 x^{2}}{- 12} = \frac{- 144}{- 12}$

Step 2.3.2.1.2

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

$x^{2} = \frac{- 144}{- 12}$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Divide $- 144$ by $- 12$ .

$x^{2} = 12$

Step 2.4

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

$x = \pm \sqrt{12}$

Step 2.5

Simplify $\pm \sqrt{12}$ .

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Step 2.5.1

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.5.1.1

Factor $4$ out of $12$ .

$x = \pm \sqrt{4 (3)}$

Step 2.5.1.2

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

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

Step 2.5.2

Pull terms out from under the radical.

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

Step 2.6

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

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Step 2.6.1

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

$x = 2 \sqrt{3}$

Step 2.6.2

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

$x = - 2 \sqrt{3}$

Step 2.6.3

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

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

Step 3

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

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Step 3.1

Substitute $2 \sqrt{3}$ in $f (x) = (x^{2} + 9) (81 - x^{2})$ to find the value of $y$ .

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Step 3.1.1

Replace the variable $x$ with $2 \sqrt{3}$ in the expression.

$f (2 \sqrt{3}) = ({(2 \sqrt{3})}^{2} + 9) (81 - {(2 \sqrt{3})}^{2})$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Apply the product rule to $2 \sqrt{3}$ .

$f (2 \sqrt{3}) = (2^{2} {\sqrt{3}}^{2} + 9) (81 - {(2 \sqrt{3})}^{2})$

Step 3.1.2.1.2

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

$f (2 \sqrt{3}) = (4 {\sqrt{3}}^{2} + 9) (81 - {(2 \sqrt{3})}^{2})$

Step 3.1.2.1.3

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

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Step 3.1.2.1.3.1

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

$f (2 \sqrt{3}) = (4 {(3^{\frac{1}{2}})}^{2} + 9) (81 - {(2 \sqrt{3})}^{2})$

Step 3.1.2.1.3.2

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

$f (2 \sqrt{3}) = (4 \cdot 3^{\frac{1}{2} \cdot 2} + 9) (81 - {(2 \sqrt{3})}^{2})$

Step 3.1.2.1.3.3

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

$f (2 \sqrt{3}) = (4 \cdot 3^{\frac{2}{2}} + 9) (81 - {(2 \sqrt{3})}^{2})$

Step 3.1.2.1.3.4

Cancel the common factor of $2$ .

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Step 3.1.2.1.3.4.1

Cancel the common factor.

$f (2 \sqrt{3}) = (4 \cdot 3^{\frac{2}{2}} + 9) (81 - {(2 \sqrt{3})}^{2})$

Step 3.1.2.1.3.4.2

Rewrite the expression.

$f (2 \sqrt{3}) = (4 \cdot 3 + 9) (81 - {(2 \sqrt{3})}^{2})$

Step 3.1.2.1.3.5

Evaluate the exponent.

$f (2 \sqrt{3}) = (4 \cdot 3 + 9) (81 - {(2 \sqrt{3})}^{2})$

Step 3.1.2.1.4

Multiply $4$ by $3$ .

$f (2 \sqrt{3}) = (12 + 9) (81 - {(2 \sqrt{3})}^{2})$

Step 3.1.2.2

Add $12$ and $9$ .

$f (2 \sqrt{3}) = 21 (81 - {(2 \sqrt{3})}^{2})$

Step 3.1.2.3

Simplify each term.

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Step 3.1.2.3.1

Apply the product rule to $2 \sqrt{3}$ .

$f (2 \sqrt{3}) = 21 (81 - (2^{2} {\sqrt{3}}^{2}))$

Step 3.1.2.3.2

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

$f (2 \sqrt{3}) = 21 (81 - (4 {\sqrt{3}}^{2}))$

Step 3.1.2.3.3

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

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Step 3.1.2.3.3.1

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

$f (2 \sqrt{3}) = 21 (81 - (4 {(3^{\frac{1}{2}})}^{2}))$

Step 3.1.2.3.3.2

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

$f (2 \sqrt{3}) = 21 (81 - (4 \cdot 3^{\frac{1}{2} \cdot 2}))$

Step 3.1.2.3.3.3

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

$f (2 \sqrt{3}) = 21 (81 - (4 \cdot 3^{\frac{2}{2}}))$

Step 3.1.2.3.3.4

Cancel the common factor of $2$ .

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Step 3.1.2.3.3.4.1

Cancel the common factor.

$f (2 \sqrt{3}) = 21 (81 - (4 \cdot 3^{\frac{2}{2}}))$

Step 3.1.2.3.3.4.2

Rewrite the expression.

$f (2 \sqrt{3}) = 21 (81 - (4 \cdot 3))$

Step 3.1.2.3.3.5

Evaluate the exponent.

$f (2 \sqrt{3}) = 21 (81 - (4 \cdot 3))$

Step 3.1.2.3.4

Multiply $- (4 \cdot 3)$ .

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Step 3.1.2.3.4.1

Multiply $4$ by $3$ .

$f (2 \sqrt{3}) = 21 (81 - 1 \cdot 12)$

Step 3.1.2.3.4.2

Multiply $- 1$ by $12$ .

$f (2 \sqrt{3}) = 21 (81 - 12)$

Step 3.1.2.4

Simplify the expression.

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Step 3.1.2.4.1

Subtract $12$ from $81$ .

$f (2 \sqrt{3}) = 21 \cdot 69$

Step 3.1.2.4.2

Multiply $21$ by $69$ .

$f (2 \sqrt{3}) = 1449$

Step 3.1.2.5

The final answer is $1449$ .

$1449$

Step 3.2

The point found by substituting $2 \sqrt{3}$ in $f (x) = (x^{2} + 9) (81 - x^{2})$ is $(2 \sqrt{3}, 1449)$ . This point can be an inflection point.

$(2 \sqrt{3}, 1449)$

Step 3.3

Substitute $- 2 \sqrt{3}$ in $f (x) = (x^{2} + 9) (81 - x^{2})$ to find the value of $y$ .

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Step 3.3.1

Replace the variable $x$ with $- 2 \sqrt{3}$ in the expression.

$f (- 2 \sqrt{3}) = ({(- 2 \sqrt{3})}^{2} + 9) (81 - {(- 2 \sqrt{3})}^{2})$

Step 3.3.2

Simplify the result.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Apply the product rule to $- 2 \sqrt{3}$ .

$f (- 2 \sqrt{3}) = ({(- 2)}^{2} {\sqrt{3}}^{2} + 9) (81 - {(- 2 \sqrt{3})}^{2})$

Step 3.3.2.1.2

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

$f (- 2 \sqrt{3}) = (4 {\sqrt{3}}^{2} + 9) (81 - {(- 2 \sqrt{3})}^{2})$

Step 3.3.2.1.3

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

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Step 3.3.2.1.3.1

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

$f (- 2 \sqrt{3}) = (4 {(3^{\frac{1}{2}})}^{2} + 9) (81 - {(- 2 \sqrt{3})}^{2})$

Step 3.3.2.1.3.2

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

$f (- 2 \sqrt{3}) = (4 \cdot 3^{\frac{1}{2} \cdot 2} + 9) (81 - {(- 2 \sqrt{3})}^{2})$

Step 3.3.2.1.3.3

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

$f (- 2 \sqrt{3}) = (4 \cdot 3^{\frac{2}{2}} + 9) (81 - {(- 2 \sqrt{3})}^{2})$

Step 3.3.2.1.3.4

Cancel the common factor of $2$ .

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Step 3.3.2.1.3.4.1

Cancel the common factor.

$f (- 2 \sqrt{3}) = (4 \cdot 3^{\frac{2}{2}} + 9) (81 - {(- 2 \sqrt{3})}^{2})$

Step 3.3.2.1.3.4.2

Rewrite the expression.

$f (- 2 \sqrt{3}) = (4 \cdot 3 + 9) (81 - {(- 2 \sqrt{3})}^{2})$

Step 3.3.2.1.3.5

Evaluate the exponent.

$f (- 2 \sqrt{3}) = (4 \cdot 3 + 9) (81 - {(- 2 \sqrt{3})}^{2})$

Step 3.3.2.1.4

Multiply $4$ by $3$ .

$f (- 2 \sqrt{3}) = (12 + 9) (81 - {(- 2 \sqrt{3})}^{2})$

Step 3.3.2.2

Add $12$ and $9$ .

$f (- 2 \sqrt{3}) = 21 (81 - {(- 2 \sqrt{3})}^{2})$

Step 3.3.2.3

Simplify each term.

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Step 3.3.2.3.1

Apply the product rule to $- 2 \sqrt{3}$ .

$f (- 2 \sqrt{3}) = 21 (81 - ({(- 2)}^{2} {\sqrt{3}}^{2}))$

Step 3.3.2.3.2

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

$f (- 2 \sqrt{3}) = 21 (81 - (4 {\sqrt{3}}^{2}))$

Step 3.3.2.3.3

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

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Step 3.3.2.3.3.1

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

$f (- 2 \sqrt{3}) = 21 (81 - (4 {(3^{\frac{1}{2}})}^{2}))$

Step 3.3.2.3.3.2

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

$f (- 2 \sqrt{3}) = 21 (81 - (4 \cdot 3^{\frac{1}{2} \cdot 2}))$

Step 3.3.2.3.3.3

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

$f (- 2 \sqrt{3}) = 21 (81 - (4 \cdot 3^{\frac{2}{2}}))$

Step 3.3.2.3.3.4

Cancel the common factor of $2$ .

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Step 3.3.2.3.3.4.1

Cancel the common factor.

$f (- 2 \sqrt{3}) = 21 (81 - (4 \cdot 3^{\frac{2}{2}}))$

Step 3.3.2.3.3.4.2

Rewrite the expression.

$f (- 2 \sqrt{3}) = 21 (81 - (4 \cdot 3))$

Step 3.3.2.3.3.5

Evaluate the exponent.

$f (- 2 \sqrt{3}) = 21 (81 - (4 \cdot 3))$

Step 3.3.2.3.4

Multiply $- (4 \cdot 3)$ .

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Step 3.3.2.3.4.1

Multiply $4$ by $3$ .

$f (- 2 \sqrt{3}) = 21 (81 - 1 \cdot 12)$

Step 3.3.2.3.4.2

Multiply $- 1$ by $12$ .

$f (- 2 \sqrt{3}) = 21 (81 - 12)$

Step 3.3.2.4

Simplify the expression.

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Step 3.3.2.4.1

Subtract $12$ from $81$ .

$f (- 2 \sqrt{3}) = 21 \cdot 69$

Step 3.3.2.4.2

Multiply $21$ by $69$ .

$f (- 2 \sqrt{3}) = 1449$

Step 3.3.2.5

The final answer is $1449$ .

$1449$

Step 3.4

The point found by substituting $- 2 \sqrt{3}$ in $f (x) = (x^{2} + 9) (81 - x^{2})$ is $(- 2 \sqrt{3}, 1449)$ . This point can be an inflection point.

$(- 2 \sqrt{3}, 1449)$

Step 3.5

Determine the points that could be inflection points.

$(2 \sqrt{3}, 1449), (- 2 \sqrt{3}, 1449)$

Step 4

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

$(- \infty, - 2 \sqrt{3}) \cup (- 2 \sqrt{3}, 2 \sqrt{3}) \cup (2 \sqrt{3}, \infty)$

Step 5

Substitute a value from the interval $(- \infty, - 2 \sqrt{3})$ into the second derivative to determine if it is increasing or decreasing.

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Step 5.1

Replace the variable $x$ with $- 3.56410161$ in the expression.

$f'' (- 3.56410161) = - 12 {(- 3.56410161)}^{2} + 144$

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 $- 3.56410161$ to the power of $2$ .

$f'' (- 3.56410161) = - 12 \cdot 12.70282032 + 144$

Step 5.2.1.2

Multiply $- 12$ by $12.70282032$ .

$f'' (- 3.56410161) = - 152.43384387 + 144$

Step 5.2.2

Add $- 152.43384387$ and $144$ .

$f'' (- 3.56410161) = - 8.43384387$

Step 5.2.3

The final answer is $- 8.43384387$ .

$- 8.43384387$

Step 5.3

At $- 3.56410161$ , the second derivative is $- 8.43384387$ . Since this is negative, the second derivative is decreasing on the interval $(- \infty, - 2 \sqrt{3})$

Decreasing on $(- \infty, - 2 \sqrt{3})$ since $f'' (x) < 0$

Step 6

Substitute a value from the interval $(- 2 \sqrt{3}, 2 \sqrt{3})$ into the second derivative to determine if it is increasing or decreasing.

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Step 6.1

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

$f'' (0) = - 12 {(0)}^{2} + 144$

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

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

$f'' (0) = - 12 \cdot 0 + 144$

Step 6.2.1.2

Multiply $- 12$ by $0$ .

$f'' (0) = 0 + 144$

Step 6.2.2

Add $0$ and $144$ .

$f'' (0) = 144$

Step 6.2.3

The final answer is $144$ .

$144$

Step 6.3

At $0$ , the second derivative is $144$ . Since this is positive, the second derivative is increasing on the interval $(- 2 \sqrt{3}, 2 \sqrt{3})$ .

Increasing on $(- 2 \sqrt{3}, 2 \sqrt{3})$ since $f'' (x) > 0$

Step 7

Substitute a value from the interval $(2 \sqrt{3}, \infty)$ into the second derivative to determine if it is increasing or decreasing.

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Step 7.1

Replace the variable $x$ with $3.56410161$ in the expression.

$f'' (3.56410161) = - 12 {(3.56410161)}^{2} + 144$

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

Raise $3.56410161$ to the power of $2$ .

$f'' (3.56410161) = - 12 \cdot 12.70282032 + 144$

Step 7.2.1.2

Multiply $- 12$ by $12.70282032$ .

$f'' (3.56410161) = - 152.43384387 + 144$

Step 7.2.2

Add $- 152.43384387$ and $144$ .

$f'' (3.56410161) = - 8.43384387$

Step 7.2.3

The final answer is $- 8.43384387$ .

$- 8.43384387$

Step 7.3

At $3.56410161$ , the second derivative is $- 8.43384387$ . Since this is negative, the second derivative is decreasing on the interval $(2 \sqrt{3}, \infty)$

Decreasing on $(2 \sqrt{3}, \infty)$ since $f'' (x) < 0$

Step 8

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 $(- 2 \sqrt{3}, 1449), (2 \sqrt{3}, 1449)$ .

$(- 2 \sqrt{3}, 1449), (2 \sqrt{3}, 1449)$

Step 9