Calculus Examples

$f (x) = \frac{1}{4} x^{4} - 3 x^{3} - \frac{81}{2} x^{2} + 729 x$

Step 1

Find the first derivative of the function.

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Step 1.1

By the Sum Rule, the derivative of $\frac{1}{4} x^{4} - 3 x^{3} - \frac{81}{2} x^{2} + 729 x$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{4} x^{4}] + \frac{d}{d x} [- 3 x^{3}] + \frac{d}{d x} [- \frac{81}{2} x^{2}] + \frac{d}{d x} [729 x]$ .

$\frac{d}{d x} [\frac{1}{4} x^{4}] + \frac{d}{d x} [- 3 x^{3}] + \frac{d}{d x} [- \frac{81}{2} x^{2}] + \frac{d}{d x} [729 x]$

Step 1.2

Evaluate $\frac{d}{d x} [\frac{1}{4} x^{4}]$ .

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Step 1.2.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{1}{4} x^{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x^{4}]$ .

$\frac{1}{4} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 3 x^{3}] + \frac{d}{d x} [- \frac{81}{2} x^{2}] + \frac{d}{d x} [729 x]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$\frac{1}{4} (4 x^{3}) + \frac{d}{d x} [- 3 x^{3}] + \frac{d}{d x} [- \frac{81}{2} x^{2}] + \frac{d}{d x} [729 x]$

Step 1.2.3

Combine $4$ and $\frac{1}{4}$ .

$\frac{4}{4} x^{3} + \frac{d}{d x} [- 3 x^{3}] + \frac{d}{d x} [- \frac{81}{2} x^{2}] + \frac{d}{d x} [729 x]$

Step 1.2.4

Combine $\frac{4}{4}$ and $x^{3}$ .

$\frac{4 x^{3}}{4} + \frac{d}{d x} [- 3 x^{3}] + \frac{d}{d x} [- \frac{81}{2} x^{2}] + \frac{d}{d x} [729 x]$

Step 1.2.5

Cancel the common factor of $4$ .

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Step 1.2.5.1

Cancel the common factor.

$\frac{4 x^{3}}{4} + \frac{d}{d x} [- 3 x^{3}] + \frac{d}{d x} [- \frac{81}{2} x^{2}] + \frac{d}{d x} [729 x]$

Step 1.2.5.2

Divide $x^{3}$ by $1$ .

$x^{3} + \frac{d}{d x} [- 3 x^{3}] + \frac{d}{d x} [- \frac{81}{2} x^{2}] + \frac{d}{d x} [729 x]$

Step 1.3

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

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Step 1.3.1

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

$x^{3} - 3 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- \frac{81}{2} x^{2}] + \frac{d}{d x} [729 x]$

Step 1.3.2

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

$x^{3} - 3 (3 x^{2}) + \frac{d}{d x} [- \frac{81}{2} x^{2}] + \frac{d}{d x} [729 x]$

Step 1.3.3

Multiply $3$ by $- 3$ .

$x^{3} - 9 x^{2} + \frac{d}{d x} [- \frac{81}{2} x^{2}] + \frac{d}{d x} [729 x]$

Step 1.4

Evaluate $\frac{d}{d x} [- \frac{81}{2} x^{2}]$ .

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Step 1.4.1

Since $- \frac{81}{2}$ is constant with respect to $x$ , the derivative of $- \frac{81}{2} x^{2}$ with respect to $x$ is $- \frac{81}{2} \frac{d}{d x} [x^{2}]$ .

$x^{3} - 9 x^{2} - \frac{81}{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [729 x]$

Step 1.4.2

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

$x^{3} - 9 x^{2} - \frac{81}{2} (2 x) + \frac{d}{d x} [729 x]$

Step 1.4.3

Multiply $2$ by $- 1$ .

$x^{3} - 9 x^{2} - 2 (\frac{81}{2}) x + \frac{d}{d x} [729 x]$

Step 1.4.4

Combine $- 2$ and $\frac{81}{2}$ .

$x^{3} - 9 x^{2} + \frac{- 2 \cdot 81}{2} x + \frac{d}{d x} [729 x]$

Step 1.4.5

Multiply $- 2$ by $81$ .

$x^{3} - 9 x^{2} + \frac{- 162}{2} x + \frac{d}{d x} [729 x]$

Step 1.4.6

Combine $\frac{- 162}{2}$ and $x$ .

$x^{3} - 9 x^{2} + \frac{- 162 x}{2} + \frac{d}{d x} [729 x]$

Step 1.4.7

Cancel the common factor of $- 162$ and $2$ .

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Step 1.4.7.1

Factor $2$ out of $- 162 x$ .

$x^{3} - 9 x^{2} + \frac{2 (- 81 x)}{2} + \frac{d}{d x} [729 x]$

Step 1.4.7.2

Cancel the common factors.

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Step 1.4.7.2.1

Factor $2$ out of $2$ .

$x^{3} - 9 x^{2} + \frac{2 (- 81 x)}{2 (1)} + \frac{d}{d x} [729 x]$

Step 1.4.7.2.2

Cancel the common factor.

$x^{3} - 9 x^{2} + \frac{2 (- 81 x)}{2 \cdot 1} + \frac{d}{d x} [729 x]$

Step 1.4.7.2.3

Rewrite the expression.

$x^{3} - 9 x^{2} + \frac{- 81 x}{1} + \frac{d}{d x} [729 x]$

Step 1.4.7.2.4

Divide $- 81 x$ by $1$ .

$x^{3} - 9 x^{2} - 81 x + \frac{d}{d x} [729 x]$

Step 1.5

Evaluate $\frac{d}{d x} [729 x]$ .

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Step 1.5.1

Since $729$ is constant with respect to $x$ , the derivative of $729 x$ with respect to $x$ is $729 \frac{d}{d x} [x]$ .

$x^{3} - 9 x^{2} - 81 x + 729 \frac{d}{d x} [x]$

Step 1.5.2

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

$x^{3} - 9 x^{2} - 81 x + 729 \cdot 1$

Step 1.5.3

Multiply $729$ by $1$ .

$x^{3} - 9 x^{2} - 81 x + 729$

Step 2

Find the second derivative of the function.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x^{3} - 9 x^{2} - 81 x + 729$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 9 x^{2}] + \frac{d}{d x} [- 81 x] + \frac{d}{d x} [729]$ .

$f'' (x) = \frac{d}{d x} (x^{3}) + \frac{d}{d x} (- 9 x^{2}) + \frac{d}{d x} (- 81 x) + \frac{d}{d x} (729)$

Step 2.1.2

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

$f'' (x) = 3 x^{2} + \frac{d}{d x} (- 9 x^{2}) + \frac{d}{d x} (- 81 x) + \frac{d}{d x} (729)$

Step 2.2

Evaluate $\frac{d}{d x} [- 9 x^{2}]$ .

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Step 2.2.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x^{2}$ with respect to $x$ is $- 9 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = 3 x^{2} - 9 \frac{d}{d x} x^{2} + \frac{d}{d x} (- 81 x) + \frac{d}{d x} (729)$

Step 2.2.2

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

$f'' (x) = 3 x^{2} - 9 (2 x) + \frac{d}{d x} (- 81 x) + \frac{d}{d x} (729)$

Step 2.2.3

Multiply $2$ by $- 9$ .

$f'' (x) = 3 x^{2} - 18 x + \frac{d}{d x} (- 81 x) + \frac{d}{d x} (729)$

Step 2.3

Evaluate $\frac{d}{d x} [- 81 x]$ .

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Step 2.3.1

Since $- 81$ is constant with respect to $x$ , the derivative of $- 81 x$ with respect to $x$ is $- 81 \frac{d}{d x} [x]$ .

$f'' (x) = 3 x^{2} - 18 x - 81 \frac{d}{d x} x + \frac{d}{d x} (729)$

Step 2.3.2

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

$f'' (x) = 3 x^{2} - 18 x - 81 \cdot 1 + \frac{d}{d x} (729)$

Step 2.3.3

Multiply $- 81$ by $1$ .

$f'' (x) = 3 x^{2} - 18 x - 81 + \frac{d}{d x} (729)$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $729$ is constant with respect to $x$ , the derivative of $729$ with respect to $x$ is $0$ .

$f'' (x) = 3 x^{2} - 18 x - 81 + 0$

Step 2.4.2

Add $3 x^{2} - 18 x - 81$ and $0$ .

$f'' (x) = 3 x^{2} - 18 x - 81$

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$x^{3} - 9 x^{2} - 81 x + 729 = 0$

Step 4

Find the first derivative.

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Step 4.1

Find the first derivative.

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Step 4.1.1

$f' (x) = \frac{d}{d x} (\frac{1}{4} x^{4}) + \frac{d}{d x} (- 3 x^{3}) + \frac{d}{d x} (- \frac{81}{2} x^{2}) + \frac{d}{d x} (729 x)$

Step 4.1.2

Evaluate $\frac{d}{d x} [\frac{1}{4} x^{4}]$ .

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Step 4.1.2.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{1}{4} x^{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x^{4}]$ .

$f' (x) = \frac{1}{4} \cdot \frac{d}{d x} (x^{4}) + \frac{d}{d x} (- 3 x^{3}) + \frac{d}{d x} (- \frac{81}{2} x^{2}) + \frac{d}{d x} (729 x)$

Step 4.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$f' (x) = \frac{1}{4} \cdot (4 x^{3}) + \frac{d}{d x} (- 3 x^{3}) + \frac{d}{d x} (- \frac{81}{2} x^{2}) + \frac{d}{d x} (729 x)$

Step 4.1.2.3

Combine $4$ and $\frac{1}{4}$ .

$f' (x) = \frac{4}{4} x^{3} + \frac{d}{d x} (- 3 x^{3}) + \frac{d}{d x} (- \frac{81}{2} x^{2}) + \frac{d}{d x} (729 x)$

Step 4.1.2.4

Combine $\frac{4}{4}$ and $x^{3}$ .

$f' (x) = \frac{4 x^{3}}{4} + \frac{d}{d x} (- 3 x^{3}) + \frac{d}{d x} (- \frac{81}{2} x^{2}) + \frac{d}{d x} (729 x)$

Step 4.1.2.5

Cancel the common factor of $4$ .

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Step 4.1.2.5.1

Cancel the common factor.

$f' (x) = \frac{4 x^{3}}{4} + \frac{d}{d x} (- 3 x^{3}) + \frac{d}{d x} (- \frac{81}{2} x^{2}) + \frac{d}{d x} (729 x)$

Step 4.1.2.5.2

Divide $x^{3}$ by $1$ .

$f' (x) = x^{3} + \frac{d}{d x} (- 3 x^{3}) + \frac{d}{d x} (- \frac{81}{2} x^{2}) + \frac{d}{d x} (729 x)$

Step 4.1.3

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

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Step 4.1.3.1

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

$f' (x) = x^{3} - 3 \frac{d}{d x} x^{3} + \frac{d}{d x} (- \frac{81}{2} x^{2}) + \frac{d}{d x} (729 x)$

Step 4.1.3.2

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

$f' (x) = x^{3} - 3 (3 x^{2}) + \frac{d}{d x} (- \frac{81}{2} x^{2}) + \frac{d}{d x} (729 x)$

Step 4.1.3.3

Multiply $3$ by $- 3$ .

$f' (x) = x^{3} - 9 x^{2} + \frac{d}{d x} (- \frac{81}{2} x^{2}) + \frac{d}{d x} (729 x)$

Step 4.1.4

Evaluate $\frac{d}{d x} [- \frac{81}{2} x^{2}]$ .

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Step 4.1.4.1

Since $- \frac{81}{2}$ is constant with respect to $x$ , the derivative of $- \frac{81}{2} x^{2}$ with respect to $x$ is $- \frac{81}{2} \frac{d}{d x} [x^{2}]$ .

$f' (x) = x^{3} - 9 x^{2} - \frac{81}{2} \cdot \frac{d}{d x} x^{2} + \frac{d}{d x} (729 x)$

Step 4.1.4.2

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

$f' (x) = x^{3} - 9 x^{2} - \frac{81}{2} \cdot (2 x) + \frac{d}{d x} (729 x)$

Step 4.1.4.3

Multiply $2$ by $- 1$ .

$f' (x) = x^{3} - 9 x^{2} - 2 (\frac{81}{2}) \cdot x + \frac{d}{d x} (729 x)$

Step 4.1.4.4

Combine $- 2$ and $\frac{81}{2}$ .

$f' (x) = x^{3} - 9 x^{2} + \frac{- 2 \cdot 81}{2} x + \frac{d}{d x} (729 x)$

Step 4.1.4.5

Multiply $- 2$ by $81$ .

$f' (x) = x^{3} - 9 x^{2} + \frac{- 162}{2} x + \frac{d}{d x} (729 x)$

Step 4.1.4.6

Combine $\frac{- 162}{2}$ and $x$ .

$f' (x) = x^{3} - 9 x^{2} + \frac{- 162 x}{2} + \frac{d}{d x} (729 x)$

Step 4.1.4.7

Cancel the common factor of $- 162$ and $2$ .

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Step 4.1.4.7.1

Factor $2$ out of $- 162 x$ .

$f' (x) = x^{3} - 9 x^{2} + \frac{2 (- 81 x)}{2} + \frac{d}{d x} (729 x)$

Step 4.1.4.7.2

Cancel the common factors.

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Step 4.1.4.7.2.1

Factor $2$ out of $2$ .

$f' (x) = x^{3} - 9 x^{2} + \frac{2 (- 81 x)}{2 (1)} + \frac{d}{d x} (729 x)$

Step 4.1.4.7.2.2

Cancel the common factor.

$f' (x) = x^{3} - 9 x^{2} + \frac{2 (- 81 x)}{2 \cdot 1} + \frac{d}{d x} (729 x)$

Step 4.1.4.7.2.3

Rewrite the expression.

$f' (x) = x^{3} - 9 x^{2} + \frac{- 81 x}{1} + \frac{d}{d x} (729 x)$

Step 4.1.4.7.2.4

Divide $- 81 x$ by $1$ .

$f' (x) = x^{3} - 9 x^{2} - 81 x + \frac{d}{d x} (729 x)$

Step 4.1.5

Evaluate $\frac{d}{d x} [729 x]$ .

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Step 4.1.5.1

Since $729$ is constant with respect to $x$ , the derivative of $729 x$ with respect to $x$ is $729 \frac{d}{d x} [x]$ .

$f' (x) = x^{3} - 9 x^{2} - 81 x + 729 \frac{d}{d x} (x)$

Step 4.1.5.2

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

$f' (x) = x^{3} - 9 x^{2} - 81 x + 729 \cdot 1$

Step 4.1.5.3

Multiply $729$ by $1$ .

$f' (x) = x^{3} - 9 x^{2} - 81 x + 729$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $x^{3} - 9 x^{2} - 81 x + 729$ .

$x^{3} - 9 x^{2} - 81 x + 729$

Step 5

Set the first derivative equal to $0$ then solve the equation $x^{3} - 9 x^{2} - 81 x + 729 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$x^{3} - 9 x^{2} - 81 x + 729 = 0$

Step 5.2

Factor the left side of the equation.

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Step 5.2.1

Factor out the greatest common factor from each group.

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Step 5.2.1.1

Group the first two terms and the last two terms.

$(x^{3} - 9 x^{2}) - 81 x + 729 = 0$

Step 5.2.1.2

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

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

Step 5.2.2

Factor the polynomial by factoring out the greatest common factor, $x - 9$ .

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

Step 5.2.3

Rewrite $81$ as $9^{2}$ .

$(x - 9) (x^{2} - 9^{2}) = 0$

Step 5.2.4

Factor.

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Step 5.2.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 9$ .

$(x - 9) ((x + 9) (x - 9)) = 0$

Step 5.2.4.2

Remove unnecessary parentheses.

$(x - 9) (x + 9) (x - 9) = 0$

Step 5.2.5

Combine exponents.

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Step 5.2.5.1

Raise $x - 9$ to the power of $1$ .

$(x - 9) (x - 9) (x + 9) = 0$

Step 5.2.5.2

Raise $x - 9$ to the power of $1$ .

$(x - 9) (x - 9) (x + 9) = 0$

Step 5.2.5.3

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

${(x - 9)}^{1 + 1} (x + 9) = 0$

Step 5.2.5.4

Add $1$ and $1$ .

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

Step 5.3

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

${(x - 9)}^{2} = 0$

$x + 9 = 0$

Step 5.4

Set ${(x - 9)}^{2}$ equal to $0$ and solve for $x$ .

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Step 5.4.1

Set ${(x - 9)}^{2}$ equal to $0$ .

${(x - 9)}^{2} = 0$

Step 5.4.2

Solve ${(x - 9)}^{2} = 0$ for $x$ .

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Step 5.4.2.1

Set the $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 5.4.2.2

Add $9$ to both sides of the equation.

$x = 9$

Step 5.5

Set $x + 9$ equal to $0$ and solve for $x$ .

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Step 5.5.1

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Step 5.5.2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 5.6

The final solution is all the values that make ${(x - 9)}^{2} (x + 9) = 0$ true.

$x = 9, - 9$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

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.

Step 7

Critical points to evaluate.

$x = 9, - 9$

Step 8

Evaluate the second derivative at $x = 9$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$3 {(9)}^{2} - 18 \cdot 9 - 81$

Step 9

Evaluate the second derivative.

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Step 9.1

Simplify each term.

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Step 9.1.1

Raise $9$ to the power of $2$ .

$3 \cdot 81 - 18 \cdot 9 - 81$

Step 9.1.2

Multiply $3$ by $81$ .

$243 - 18 \cdot 9 - 81$

Step 9.1.3

Multiply $- 18$ by $9$ .

$243 - 162 - 81$

Step 9.2

Simplify by subtracting numbers.

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Step 9.2.1

Subtract $162$ from $243$ .

$81 - 81$

Step 9.2.2

Subtract $81$ from $81$ .

$0$

Step 10

Since there is at least one point with $0$ or undefined second derivative, apply the first derivative test.

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Step 10.1

Split $(- \infty, \infty)$ into separate intervals around the $x$ values that make the first derivative $0$ or undefined.

$(- \infty, - 9) \cup (- 9, 9) \cup (9, \infty)$

Step 10.2

Substitute any number, such as $- 12$ , from the interval $(- \infty, - 9)$ in the first derivative $x^{3} - 9 x^{2} - 81 x + 729$ to check if the result is negative or positive.

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Step 10.2.1

Replace the variable $x$ with $- 12$ in the expression.

$f' (- 12) = {(- 12)}^{3} - 9 {(- 12)}^{2} - 81 \cdot - 12 + 729$

Step 10.2.2

Simplify the result.

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Step 10.2.2.1

Simplify each term.

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Step 10.2.2.1.1

Raise $- 12$ to the power of $3$ .

$f' (- 12) = - 1728 - 9 {(- 12)}^{2} - 81 \cdot - 12 + 729$

Step 10.2.2.1.2

Raise $- 12$ to the power of $2$ .

$f' (- 12) = - 1728 - 9 \cdot 144 - 81 \cdot - 12 + 729$

Step 10.2.2.1.3

Multiply $- 9$ by $144$ .

$f' (- 12) = - 1728 - 1296 - 81 \cdot - 12 + 729$

Step 10.2.2.1.4

Multiply $- 81$ by $- 12$ .

$f' (- 12) = - 1728 - 1296 + 972 + 729$

Step 10.2.2.2

Simplify by adding and subtracting.

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Step 10.2.2.2.1

Subtract $1296$ from $- 1728$ .

$f' (- 12) = - 3024 + 972 + 729$

Step 10.2.2.2.2

Add $- 3024$ and $972$ .

$f' (- 12) = - 2052 + 729$

Step 10.2.2.2.3

Add $- 2052$ and $729$ .

$f' (- 12) = - 1323$

Step 10.2.2.3

The final answer is $- 1323$ .

$- 1323$

Step 10.3

Substitute any number, such as $0$ , from the interval $(- 9, 9)$ in the first derivative $x^{3} - 9 x^{2} - 81 x + 729$ to check if the result is negative or positive.

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Step 10.3.1

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

$f' (0) = {(0)}^{3} - 9 {(0)}^{2} - 81 \cdot 0 + 729$

Step 10.3.2

Simplify the result.

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Step 10.3.2.1

Simplify each term.

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Step 10.3.2.1.1

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

$f' (0) = 0 - 9 {(0)}^{2} - 81 \cdot 0 + 729$

Step 10.3.2.1.2

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

$f' (0) = 0 - 9 \cdot 0 - 81 \cdot 0 + 729$

Step 10.3.2.1.3

Multiply $- 9$ by $0$ .

$f' (0) = 0 + 0 - 81 \cdot 0 + 729$

Step 10.3.2.1.4

Multiply $- 81$ by $0$ .

$f' (0) = 0 + 0 + 0 + 729$

Step 10.3.2.2

Simplify by adding numbers.

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Step 10.3.2.2.1

Add $0$ and $0$ .

$f' (0) = 0 + 0 + 729$

Step 10.3.2.2.2

Add $0$ and $0$ .

$f' (0) = 0 + 729$

Step 10.3.2.2.3

Add $0$ and $729$ .

$f' (0) = 729$

Step 10.3.2.3

The final answer is $729$ .

$729$

Step 10.4

Substitute any number, such as $12$ , from the interval $(9, \infty)$ in the first derivative $x^{3} - 9 x^{2} - 81 x + 729$ to check if the result is negative or positive.

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Step 10.4.1

Replace the variable $x$ with $12$ in the expression.

$f' (12) = {(12)}^{3} - 9 {(12)}^{2} - 81 \cdot 12 + 729$

Step 10.4.2

Simplify the result.

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Step 10.4.2.1

Simplify each term.

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Step 10.4.2.1.1

Raise $12$ to the power of $3$ .

$f' (12) = 1728 - 9 {(12)}^{2} - 81 \cdot 12 + 729$

Step 10.4.2.1.2

Raise $12$ to the power of $2$ .

$f' (12) = 1728 - 9 \cdot 144 - 81 \cdot 12 + 729$

Step 10.4.2.1.3

Multiply $- 9$ by $144$ .

$f' (12) = 1728 - 1296 - 81 \cdot 12 + 729$

Step 10.4.2.1.4

Multiply $- 81$ by $12$ .

$f' (12) = 1728 - 1296 - 972 + 729$

Step 10.4.2.2

Simplify by adding and subtracting.

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Step 10.4.2.2.1

Subtract $1296$ from $1728$ .

$f' (12) = 432 - 972 + 729$

Step 10.4.2.2.2

Subtract $972$ from $432$ .

$f' (12) = - 540 + 729$

Step 10.4.2.2.3

Add $- 540$ and $729$ .

$f' (12) = 189$

Step 10.4.2.3

The final answer is $189$ .

$189$

Step 10.5

Since the first derivative changed signs from negative to positive around $x = - 9$ , then $x = - 9$ is a local minimum.

$x = - 9$ is a local minimum

Step 10.6

Since the first derivative did not change signs around $x = 9$ , this is not a local maximum or minimum.

Not a local maximum or minimum

Step 10.7

These are the local extrema for $f (x) = \frac{1}{4} x^{4} - 3 x^{3} - \frac{81}{2} x^{2} + 729 x$ .

$x = - 9$ is a local minimum

Step 11