Calculus Examples

$f (x) = x^{2} - 20 x + 110 + \frac{1014}{x}$

Step 1

Find the first derivative of the function.

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Step 1.1

Differentiate.

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Step 1.1.1

By the Sum Rule, the derivative of $x^{2} - 20 x + 110 + \frac{1014}{x}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [110] + \frac{d}{d x} [\frac{1014}{x}]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [110] + \frac{d}{d x} [\frac{1014}{x}]$

Step 1.1.2

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

$2 x + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [110] + \frac{d}{d x} [\frac{1014}{x}]$

Step 1.2

Evaluate $\frac{d}{d x} [- 20 x]$ .

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Step 1.2.1

Since $- 20$ is constant with respect to $x$ , the derivative of $- 20 x$ with respect to $x$ is $- 20 \frac{d}{d x} [x]$ .

$2 x - 20 \frac{d}{d x} [x] + \frac{d}{d x} [110] + \frac{d}{d x} [\frac{1014}{x}]$

Step 1.2.2

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

$2 x - 20 \cdot 1 + \frac{d}{d x} [110] + \frac{d}{d x} [\frac{1014}{x}]$

Step 1.2.3

Multiply $- 20$ by $1$ .

$2 x - 20 + \frac{d}{d x} [110] + \frac{d}{d x} [\frac{1014}{x}]$

Step 1.3

Since $110$ is constant with respect to $x$ , the derivative of $110$ with respect to $x$ is $0$ .

$2 x - 20 + 0 + \frac{d}{d x} [\frac{1014}{x}]$

Step 1.4

Evaluate $\frac{d}{d x} [\frac{1014}{x}]$ .

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Step 1.4.1

Since $1014$ is constant with respect to $x$ , the derivative of $\frac{1014}{x}$ with respect to $x$ is $1014 \frac{d}{d x} [\frac{1}{x}]$ .

$2 x - 20 + 0 + 1014 \frac{d}{d x} [\frac{1}{x}]$

Step 1.4.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$2 x - 20 + 0 + 1014 \frac{d}{d x} [x^{- 1}]$

Step 1.4.3

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

$2 x - 20 + 0 + 1014 (- x^{- 2})$

Step 1.4.4

Multiply $- 1$ by $1014$ .

$2 x - 20 + 0 - 1014 x^{- 2}$

Step 1.5

Simplify.

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Step 1.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 x - 20 + 0 - 1014 \frac{1}{x^{2}}$

Step 1.5.2

Combine terms.

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Step 1.5.2.1

Add $2 x - 20$ and $0$ .

$2 x - 20 - 1014 \frac{1}{x^{2}}$

Step 1.5.2.2

Combine $- 1014$ and $\frac{1}{x^{2}}$ .

$2 x - 20 + \frac{- 1014}{x^{2}}$

Step 1.5.2.3

Move the negative in front of the fraction.

$2 x - 20 - \frac{1014}{x^{2}}$

Step 1.5.3

Reorder terms.

$2 x - \frac{1014}{x^{2}} - 20$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $2 x - \frac{1014}{x^{2}} - 20$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{1014}{x^{2}}] + \frac{d}{d x} [- 20]$ .

$f'' (x) = \frac{d}{d x} (2 x) + \frac{d}{d x} (- \frac{1014}{x^{2}}) + \frac{d}{d x} (- 20)$

Step 2.2

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

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Step 2.2.1

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

$f'' (x) = 2 \frac{d}{d x} (x) + \frac{d}{d x} (- \frac{1014}{x^{2}}) + \frac{d}{d x} (- 20)$

Step 2.2.2

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

$f'' (x) = 2 \cdot 1 + \frac{d}{d x} (- \frac{1014}{x^{2}}) + \frac{d}{d x} (- 20)$

Step 2.2.3

Multiply $2$ by $1$ .

$f'' (x) = 2 + \frac{d}{d x} (- \frac{1014}{x^{2}}) + \frac{d}{d x} (- 20)$

Step 2.3

Evaluate $\frac{d}{d x} [- \frac{1014}{x^{2}}]$ .

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Step 2.3.1

Since $- 1014$ is constant with respect to $x$ , the derivative of $- \frac{1014}{x^{2}}$ with respect to $x$ is $- 1014 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$f'' (x) = 2 - 1014 \frac{d}{d x} \frac{1}{x^{2}} + \frac{d}{d x} (- 20)$

Step 2.3.2

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

$f'' (x) = 2 - 1014 \frac{d}{d x} {(x^{2})}^{- 1} + \frac{d}{d x} (- 20)$

Step 2.3.3

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

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Step 2.3.3.1

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

$f'' (x) = 2 - 1014 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{2})) + \frac{d}{d x} (- 20)$

Step 2.3.3.2

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

$f'' (x) = 2 - 1014 (- u^{- 2} \frac{d}{d x} x^{2}) + \frac{d}{d x} (- 20)$

Step 2.3.3.3

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

$f'' (x) = 2 - 1014 (- {(x^{2})}^{- 2} \frac{d}{d x} x^{2}) + \frac{d}{d x} (- 20)$

Step 2.3.4

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

$f'' (x) = 2 - 1014 (- {(x^{2})}^{- 2} (2 x)) + \frac{d}{d x} (- 20)$

Step 2.3.5

Multiply the exponents in ${(x^{2})}^{- 2}$ .

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Step 2.3.5.1

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

$f'' (x) = 2 - 1014 (- x^{2 \cdot - 2} (2 x)) + \frac{d}{d x} (- 20)$

Step 2.3.5.2

Multiply $2$ by $- 2$ .

$f'' (x) = 2 - 1014 (- x^{- 4} (2 x)) + \frac{d}{d x} (- 20)$

Step 2.3.6

Multiply $2$ by $- 1$ .

$f'' (x) = 2 - 1014 (- 2 x^{- 4} x) + \frac{d}{d x} (- 20)$

Step 2.3.7

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

$f'' (x) = 2 - 1014 (- 2 (x \cdot x^{- 4})) + \frac{d}{d x} (- 20)$

Step 2.3.8

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

$f'' (x) = 2 - 1014 (- 2 x^{1 - 4}) + \frac{d}{d x} (- 20)$

Step 2.3.9

Subtract $4$ from $1$ .

$f'' (x) = 2 - 1014 (- 2 x^{- 3}) + \frac{d}{d x} (- 20)$

Step 2.3.10

Multiply $- 2$ by $- 1014$ .

$f'' (x) = 2 + 2028 x^{- 3} + \frac{d}{d x} (- 20)$

Step 2.4

Since $- 20$ is constant with respect to $x$ , the derivative of $- 20$ with respect to $x$ is $0$ .

$f'' (x) = 2 + 2028 x^{- 3} + 0$

Step 2.5

Simplify.

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Step 2.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = 2 + 2028 (\frac{1}{x^{3}}) + 0$

Step 2.5.2

Combine terms.

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Step 2.5.2.1

Combine $2028$ and $\frac{1}{x^{3}}$ .

$f'' (x) = 2 + \frac{2028}{x^{3}} + 0$

Step 2.5.2.2

Add $2 + \frac{2028}{x^{3}}$ and $0$ .

$f'' (x) = 2 + \frac{2028}{x^{3}}$

Step 2.5.3

Reorder terms.

$f'' (x) = \frac{2028}{x^{3}} + 2$

Step 3

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

$2 x - \frac{1014}{x^{2}} - 20 = 0$

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

Differentiate.

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Step 4.1.1.1

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [110] + \frac{d}{d x} [\frac{1014}{x}]$

Step 4.1.1.2

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

$2 x + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [110] + \frac{d}{d x} [\frac{1014}{x}]$

Step 4.1.2

Evaluate $\frac{d}{d x} [- 20 x]$ .

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Step 4.1.2.1

Since $- 20$ is constant with respect to $x$ , the derivative of $- 20 x$ with respect to $x$ is $- 20 \frac{d}{d x} [x]$ .

$2 x - 20 \frac{d}{d x} [x] + \frac{d}{d x} [110] + \frac{d}{d x} [\frac{1014}{x}]$

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 $n = 1$ .

$2 x - 20 \cdot 1 + \frac{d}{d x} [110] + \frac{d}{d x} [\frac{1014}{x}]$

Step 4.1.2.3

Multiply $- 20$ by $1$ .

$2 x - 20 + \frac{d}{d x} [110] + \frac{d}{d x} [\frac{1014}{x}]$

Step 4.1.3

Since $110$ is constant with respect to $x$ , the derivative of $110$ with respect to $x$ is $0$ .

$2 x - 20 + 0 + \frac{d}{d x} [\frac{1014}{x}]$

Step 4.1.4

Evaluate $\frac{d}{d x} [\frac{1014}{x}]$ .

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Step 4.1.4.1

Since $1014$ is constant with respect to $x$ , the derivative of $\frac{1014}{x}$ with respect to $x$ is $1014 \frac{d}{d x} [\frac{1}{x}]$ .

$2 x - 20 + 0 + 1014 \frac{d}{d x} [\frac{1}{x}]$

Step 4.1.4.2

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$2 x - 20 + 0 + 1014 \frac{d}{d x} [x^{- 1}]$

Step 4.1.4.3

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

$2 x - 20 + 0 + 1014 (- x^{- 2})$

Step 4.1.4.4

Multiply $- 1$ by $1014$ .

$2 x - 20 + 0 - 1014 x^{- 2}$

Step 4.1.5

Simplify.

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Step 4.1.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 x - 20 + 0 - 1014 \frac{1}{x^{2}}$

Step 4.1.5.2

Combine terms.

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Step 4.1.5.2.1

Add $2 x - 20$ and $0$ .

$2 x - 20 - 1014 \frac{1}{x^{2}}$

Step 4.1.5.2.2

Combine $- 1014$ and $\frac{1}{x^{2}}$ .

$2 x - 20 + \frac{- 1014}{x^{2}}$

Step 4.1.5.2.3

Move the negative in front of the fraction.

$2 x - 20 - \frac{1014}{x^{2}}$

Step 4.1.5.3

Reorder terms.

$f' (x) = 2 x - \frac{1014}{x^{2}} - 20$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $2 x - \frac{1014}{x^{2}} - 20$ .

$2 x - \frac{1014}{x^{2}} - 20$

Step 5

Set the first derivative equal to $0$ then solve the equation $2 x - \frac{1014}{x^{2}} - 20 = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$2 x - \frac{1014}{x^{2}} - 20 = 0$

Step 5.2

Find the LCD of the terms in the equation.

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Step 5.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x^{2}, 1, 1$

Step 5.2.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 5.3

Multiply each term in $2 x - \frac{1014}{x^{2}} - 20 = 0$ by $x^{2}$ to eliminate the fractions.

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Step 5.3.1

Multiply each term in $2 x - \frac{1014}{x^{2}} - 20 = 0$ by $x^{2}$ .

$2 x \cdot x^{2} - \frac{1014}{x^{2}} x^{2} - 20 x^{2} = 0 x^{2}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Simplify each term.

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Step 5.3.2.1.1

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

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Step 5.3.2.1.1.1

Move $x^{2}$ .

$2 (x^{2} x) - \frac{1014}{x^{2}} x^{2} - 20 x^{2} = 0 x^{2}$

Step 5.3.2.1.1.2

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

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Step 5.3.2.1.1.2.1

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

$2 (x^{2} x^{1}) - \frac{1014}{x^{2}} x^{2} - 20 x^{2} = 0 x^{2}$

Step 5.3.2.1.1.2.2

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

$2 x^{2 + 1} - \frac{1014}{x^{2}} x^{2} - 20 x^{2} = 0 x^{2}$

Step 5.3.2.1.1.3

Add $2$ and $1$ .

$2 x^{3} - \frac{1014}{x^{2}} x^{2} - 20 x^{2} = 0 x^{2}$

Step 5.3.2.1.2

Cancel the common factor of $x^{2}$ .

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Step 5.3.2.1.2.1

Move the leading negative in $- \frac{1014}{x^{2}}$ into the numerator.

$2 x^{3} + \frac{- 1014}{x^{2}} x^{2} - 20 x^{2} = 0 x^{2}$

Step 5.3.2.1.2.2

Cancel the common factor.

$2 x^{3} + \frac{- 1014}{x^{2}} x^{2} - 20 x^{2} = 0 x^{2}$

Step 5.3.2.1.2.3

Rewrite the expression.

$2 x^{3} - 1014 - 20 x^{2} = 0 x^{2}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Multiply $0$ by $x^{2}$ .

$2 x^{3} - 1014 - 20 x^{2} = 0$

Step 5.4

Solve the equation.

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Step 5.4.1

Factor the left side of the equation.

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Step 5.4.1.1

Factor $2$ out of $2 x^{3} - 1014 - 20 x^{2}$ .

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Step 5.4.1.1.1

Factor $2$ out of $2 x^{3}$ .

$2 (x^{3}) - 1014 - 20 x^{2} = 0$

Step 5.4.1.1.2

Factor $2$ out of $- 1014$ .

$2 x^{3} + 2 \cdot - 507 - 20 x^{2} = 0$

Step 5.4.1.1.3

Factor $2$ out of $- 20 x^{2}$ .

$2 x^{3} + 2 \cdot - 507 + 2 (- 10 x^{2}) = 0$

Step 5.4.1.1.4

Factor $2$ out of $2 x^{3} + 2 \cdot - 507$ .

$2 (x^{3} - 507) + 2 (- 10 x^{2}) = 0$

Step 5.4.1.1.5

Factor $2$ out of $2 (x^{3} - 507) + 2 (- 10 x^{2})$ .

$2 (x^{3} - 507 - 10 x^{2}) = 0$

Step 5.4.1.2

Reorder terms.

$2 (x^{3} - 10 x^{2} - 507) = 0$

Step 5.4.1.3

Factor.

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Step 5.4.1.3.1

Factor $x^{3} - 10 x^{2} - 507$ using the rational roots test.

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Step 5.4.1.3.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 507, \pm 3, \pm 169, \pm 13, \pm 39$

$q = \pm 1$

Step 5.4.1.3.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 507, \pm 3, \pm 169, \pm 13, \pm 39$

Step 5.4.1.3.1.3

Substitute $13$ and simplify the expression. In this case, the expression is equal to $0$ so $13$ is a root of the polynomial.

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Step 5.4.1.3.1.3.1

Substitute $13$ into the polynomial.

$13^{3} - 10 \cdot 13^{2} - 507$

Step 5.4.1.3.1.3.2

Raise $13$ to the power of $3$ .

$2197 - 10 \cdot 13^{2} - 507$

Step 5.4.1.3.1.3.3

Raise $13$ to the power of $2$ .

$2197 - 10 \cdot 169 - 507$

Step 5.4.1.3.1.3.4

Multiply $- 10$ by $169$ .

$2197 - 1690 - 507$

Step 5.4.1.3.1.3.5

Subtract $1690$ from $2197$ .

$507 - 507$

Step 5.4.1.3.1.3.6

Subtract $507$ from $507$ .

$0$

Step 5.4.1.3.1.4

Since $13$ is a known root, divide the polynomial by $x - 13$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - 10 x^{2} - 507}{x - 13}$

Step 5.4.1.3.1.5

Divide $x^{3} - 10 x^{2} - 507$ by $x - 13$ .

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Step 5.4.1.3.1.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x$

$13$

$x^{3}$

$10 x^{2}$

$0 x$

$507$

Step 5.4.1.3.1.5.2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $x$ .

					$x^{2}$
	$x$	-	$13$		$x^{3}$	-	$10 x^{2}$	+	$0 x$	-	$507$

Step 5.4.1.3.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$13$		$x^{3}$	-	$10 x^{2}$	+	$0 x$	-	$507$
			+	$x^{3}$	-	$13 x^{2}$

Step 5.4.1.3.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} - 13 x^{2}$

				$x^{2}$
$x$	-	$13$		$x^{3}$	-	$10 x^{2}$	+	$0 x$	-	$507$
			-	$x^{3}$	+	$13 x^{2}$

Step 5.4.1.3.1.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$
$x$	-	$13$		$x^{3}$	-	$10 x^{2}$	+	$0 x$	-	$507$
			-	$x^{3}$	+	$13 x^{2}$
					+	$3 x^{2}$

Step 5.4.1.3.1.5.6

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$
$x$	-	$13$		$x^{3}$	-	$10 x^{2}$	+	$0 x$	-	$507$
			-	$x^{3}$	+	$13 x^{2}$
					+	$3 x^{2}$	+	$0 x$

Step 5.4.1.3.1.5.7

Divide the highest order term in the dividend $3 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$3 x$
$x$	-	$13$		$x^{3}$	-	$10 x^{2}$	+	$0 x$	-	$507$
			-	$x^{3}$	+	$13 x^{2}$
					+	$3 x^{2}$	+	$0 x$

Step 5.4.1.3.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$
$x$	-	$13$		$x^{3}$	-	$10 x^{2}$	+	$0 x$	-	$507$
			-	$x^{3}$	+	$13 x^{2}$
					+	$3 x^{2}$	+	$0 x$
					+	$3 x^{2}$	-	$39 x$

Step 5.4.1.3.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{2} - 39 x$

				$x^{2}$	+	$3 x$
$x$	-	$13$		$x^{3}$	-	$10 x^{2}$	+	$0 x$	-	$507$
			-	$x^{3}$	+	$13 x^{2}$
					+	$3 x^{2}$	+	$0 x$
					-	$3 x^{2}$	+	$39 x$

Step 5.4.1.3.1.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$3 x$
$x$	-	$13$		$x^{3}$	-	$10 x^{2}$	+	$0 x$	-	$507$
			-	$x^{3}$	+	$13 x^{2}$
					+	$3 x^{2}$	+	$0 x$
					-	$3 x^{2}$	+	$39 x$
							+	$39 x$

Step 5.4.1.3.1.5.11

Pull the next terms from the original dividend down into the current dividend.

				$x^{2}$	+	$3 x$
$x$	-	$13$		$x^{3}$	-	$10 x^{2}$	+	$0 x$	-	$507$
			-	$x^{3}$	+	$13 x^{2}$
					+	$3 x^{2}$	+	$0 x$
					-	$3 x^{2}$	+	$39 x$
							+	$39 x$	-	$507$

Step 5.4.1.3.1.5.12

Divide the highest order term in the dividend $39 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$3 x$	+	$39$
$x$	-	$13$		$x^{3}$	-	$10 x^{2}$	+	$0 x$	-	$507$
			-	$x^{3}$	+	$13 x^{2}$
					+	$3 x^{2}$	+	$0 x$
					-	$3 x^{2}$	+	$39 x$
							+	$39 x$	-	$507$

Step 5.4.1.3.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$3 x$	+	$39$
$x$	-	$13$		$x^{3}$	-	$10 x^{2}$	+	$0 x$	-	$507$
			-	$x^{3}$	+	$13 x^{2}$
					+	$3 x^{2}$	+	$0 x$
					-	$3 x^{2}$	+	$39 x$
							+	$39 x$	-	$507$
							+	$39 x$	-	$507$

Step 5.4.1.3.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $39 x - 507$

				$x^{2}$	+	$3 x$	+	$39$
$x$	-	$13$		$x^{3}$	-	$10 x^{2}$	+	$0 x$	-	$507$
			-	$x^{3}$	+	$13 x^{2}$
					+	$3 x^{2}$	+	$0 x$
					-	$3 x^{2}$	+	$39 x$
							+	$39 x$	-	$507$
							-	$39 x$	+	$507$

Step 5.4.1.3.1.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$x^{2}$	+	$3 x$	+	$39$
$x$	-	$13$		$x^{3}$	-	$10 x^{2}$	+	$0 x$	-	$507$
			-	$x^{3}$	+	$13 x^{2}$
					+	$3 x^{2}$	+	$0 x$
					-	$3 x^{2}$	+	$39 x$
							+	$39 x$	-	$507$
							-	$39 x$	+	$507$
										$0$

Step 5.4.1.3.1.5.16

Since the remander is $0$ , the final answer is the quotient.

$x^{2} + 3 x + 39$

Step 5.4.1.3.1.6

Write $x^{3} - 10 x^{2} - 507$ as a set of factors.

$2 ((x - 13) (x^{2} + 3 x + 39)) = 0$

Step 5.4.1.3.2

Remove unnecessary parentheses.

$2 (x - 13) (x^{2} + 3 x + 39) = 0$

Step 5.4.2

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

$x - 13 = 0$

$x^{2} + 3 x + 39 = 0$

Step 5.4.3

Set $x - 13$ equal to $0$ and solve for $x$ .

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Step 5.4.3.1

Set $x - 13$ equal to $0$ .

$x - 13 = 0$

Step 5.4.3.2

Add $13$ to both sides of the equation.

$x = 13$

Step 5.4.4

Set $x^{2} + 3 x + 39$ equal to $0$ and solve for $x$ .

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Step 5.4.4.1

Set $x^{2} + 3 x + 39$ equal to $0$ .

$x^{2} + 3 x + 39 = 0$

Step 5.4.4.2

Solve $x^{2} + 3 x + 39 = 0$ for $x$ .

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Step 5.4.4.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5.4.4.2.2

Substitute the values $a = 1$ , $b = 3$ , and $c = 39$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot 39)}}{2 \cdot 1}$

Step 5.4.4.2.3

Simplify.

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Step 5.4.4.2.3.1

Simplify the numerator.

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Step 5.4.4.2.3.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 39}}{2 \cdot 1}$

Step 5.4.4.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 39$ .

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Step 5.4.4.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 39}}{2 \cdot 1}$

Step 5.4.4.2.3.1.2.2

Multiply $- 4$ by $39$ .

$x = \frac{- 3 \pm \sqrt{9 - 156}}{2 \cdot 1}$

Step 5.4.4.2.3.1.3

Subtract $156$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 147}}{2 \cdot 1}$

Step 5.4.4.2.3.1.4

Rewrite $- 147$ as $- 1 (147)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 147}}{2 \cdot 1}$

Step 5.4.4.2.3.1.5

Rewrite $\sqrt{- 1 (147)}$ as $\sqrt{- 1} \cdot \sqrt{147}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{147}}{2 \cdot 1}$

Step 5.4.4.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 3 \pm i \cdot \sqrt{147}}{2 \cdot 1}$

Step 5.4.4.2.3.1.7

Rewrite $147$ as $7^{2} \cdot 3$ .

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Step 5.4.4.2.3.1.7.1

Factor $49$ out of $147$ .

$x = \frac{- 3 \pm i \cdot \sqrt{49 (3)}}{2 \cdot 1}$

Step 5.4.4.2.3.1.7.2

Rewrite $49$ as $7^{2}$ .

$x = \frac{- 3 \pm i \cdot \sqrt{7^{2} \cdot 3}}{2 \cdot 1}$

Step 5.4.4.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 3 \pm i \cdot (7 \sqrt{3})}{2 \cdot 1}$

Step 5.4.4.2.3.1.9

Move $7$ to the left of $i$ .

$x = \frac{- 3 \pm 7 i \sqrt{3}}{2 \cdot 1}$

Step 5.4.4.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm 7 i \sqrt{3}}{2}$

Step 5.4.4.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 5.4.4.2.4.1

Simplify the numerator.

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Step 5.4.4.2.4.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 39}}{2 \cdot 1}$

Step 5.4.4.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 39$ .

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Step 5.4.4.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 39}}{2 \cdot 1}$

Step 5.4.4.2.4.1.2.2

Multiply $- 4$ by $39$ .

$x = \frac{- 3 \pm \sqrt{9 - 156}}{2 \cdot 1}$

Step 5.4.4.2.4.1.3

Subtract $156$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 147}}{2 \cdot 1}$

Step 5.4.4.2.4.1.4

Rewrite $- 147$ as $- 1 (147)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 147}}{2 \cdot 1}$

Step 5.4.4.2.4.1.5

Rewrite $\sqrt{- 1 (147)}$ as $\sqrt{- 1} \cdot \sqrt{147}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{147}}{2 \cdot 1}$

Step 5.4.4.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 3 \pm i \cdot \sqrt{147}}{2 \cdot 1}$

Step 5.4.4.2.4.1.7

Rewrite $147$ as $7^{2} \cdot 3$ .

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Step 5.4.4.2.4.1.7.1

Factor $49$ out of $147$ .

$x = \frac{- 3 \pm i \cdot \sqrt{49 (3)}}{2 \cdot 1}$

Step 5.4.4.2.4.1.7.2

Rewrite $49$ as $7^{2}$ .

$x = \frac{- 3 \pm i \cdot \sqrt{7^{2} \cdot 3}}{2 \cdot 1}$

Step 5.4.4.2.4.1.8

Pull terms out from under the radical.

$x = \frac{- 3 \pm i \cdot (7 \sqrt{3})}{2 \cdot 1}$

Step 5.4.4.2.4.1.9

Move $7$ to the left of $i$ .

$x = \frac{- 3 \pm 7 i \sqrt{3}}{2 \cdot 1}$

Step 5.4.4.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm 7 i \sqrt{3}}{2}$

Step 5.4.4.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 3 + 7 i \sqrt{3}}{2}$

Step 5.4.4.2.4.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + 7 i \sqrt{3}}{2}$

Step 5.4.4.2.4.5

Factor $- 1$ out of $7 i \sqrt{3}$ .

$x = \frac{- 1 \cdot 3 - (- 7 i \sqrt{3})}{2}$

Step 5.4.4.2.4.6

Factor $- 1$ out of $- 1 (3) - (- 7 i \sqrt{3})$ .

$x = \frac{- 1 (3 - 7 i \sqrt{3})}{2}$

Step 5.4.4.2.4.7

Move the negative in front of the fraction.

$x = - \frac{3 - 7 i \sqrt{3}}{2}$

Step 5.4.4.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 5.4.4.2.5.1

Simplify the numerator.

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Step 5.4.4.2.5.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 39}}{2 \cdot 1}$

Step 5.4.4.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 39$ .

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Step 5.4.4.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 39}}{2 \cdot 1}$

Step 5.4.4.2.5.1.2.2

Multiply $- 4$ by $39$ .

$x = \frac{- 3 \pm \sqrt{9 - 156}}{2 \cdot 1}$

Step 5.4.4.2.5.1.3

Subtract $156$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 147}}{2 \cdot 1}$

Step 5.4.4.2.5.1.4

Rewrite $- 147$ as $- 1 (147)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 147}}{2 \cdot 1}$

Step 5.4.4.2.5.1.5

Rewrite $\sqrt{- 1 (147)}$ as $\sqrt{- 1} \cdot \sqrt{147}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{147}}{2 \cdot 1}$

Step 5.4.4.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 3 \pm i \cdot \sqrt{147}}{2 \cdot 1}$

Step 5.4.4.2.5.1.7

Rewrite $147$ as $7^{2} \cdot 3$ .

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Step 5.4.4.2.5.1.7.1

Factor $49$ out of $147$ .

$x = \frac{- 3 \pm i \cdot \sqrt{49 (3)}}{2 \cdot 1}$

Step 5.4.4.2.5.1.7.2

Rewrite $49$ as $7^{2}$ .

$x = \frac{- 3 \pm i \cdot \sqrt{7^{2} \cdot 3}}{2 \cdot 1}$

Step 5.4.4.2.5.1.8

Pull terms out from under the radical.

$x = \frac{- 3 \pm i \cdot (7 \sqrt{3})}{2 \cdot 1}$

Step 5.4.4.2.5.1.9

Move $7$ to the left of $i$ .

$x = \frac{- 3 \pm 7 i \sqrt{3}}{2 \cdot 1}$

Step 5.4.4.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm 7 i \sqrt{3}}{2}$

Step 5.4.4.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 3 - 7 i \sqrt{3}}{2}$

Step 5.4.4.2.5.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - 7 i \sqrt{3}}{2}$

Step 5.4.4.2.5.5

Factor $- 1$ out of $- 7 i \sqrt{3}$ .

$x = \frac{- 1 \cdot 3 - (7 i \sqrt{3})}{2}$

Step 5.4.4.2.5.6

Factor $- 1$ out of $- 1 (3) - (7 i \sqrt{3})$ .

$x = \frac{- 1 (3 + 7 i \sqrt{3})}{2}$

Step 5.4.4.2.5.7

Move the negative in front of the fraction.

$x = - \frac{3 + 7 i \sqrt{3}}{2}$

Step 5.4.4.2.6

The final answer is the combination of both solutions.

$x = - \frac{3 - 7 i \sqrt{3}}{2}, - \frac{3 + 7 i \sqrt{3}}{2}$

Step 5.4.5

The final solution is all the values that make $2 (x - 13) (x^{2} + 3 x + 39) = 0$ true.

$x = 13, - \frac{3 - 7 i \sqrt{3}}{2}, - \frac{3 + 7 i \sqrt{3}}{2}$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{1014}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 6.2

Solve for $x$ .

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Step 6.2.1

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

$x = \pm \sqrt{0}$

Step 6.2.2

Simplify $\pm \sqrt{0}$ .

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Step 6.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 6.2.2.2

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

$x = \pm 0$

Step 6.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 7

Critical points to evaluate.

$x = 13$

Step 8

Evaluate the second derivative at $x = 13$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{2028}{{(13)}^{3}} + 2$

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 $13$ to the power of $3$ .

$\frac{2028}{2197} + 2$

Step 9.1.2

Cancel the common factor of $2028$ and $2197$ .

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Step 9.1.2.1

Factor $169$ out of $2028$ .

$\frac{169 (12)}{2197} + 2$

Step 9.1.2.2

Cancel the common factors.

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Step 9.1.2.2.1

Factor $169$ out of $2197$ .

$\frac{169 \cdot 12}{169 \cdot 13} + 2$

Step 9.1.2.2.2

Cancel the common factor.

$\frac{169 \cdot 12}{169 \cdot 13} + 2$

Step 9.1.2.2.3

Rewrite the expression.

$\frac{12}{13} + 2$

Step 9.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$\frac{12}{13} + 2 \cdot \frac{13}{13}$

Step 9.3

Combine $2$ and $\frac{13}{13}$ .

$\frac{12}{13} + \frac{2 \cdot 13}{13}$

Step 9.4

Combine the numerators over the common denominator.

$\frac{12 + 2 \cdot 13}{13}$

Step 9.5

Simplify the numerator.

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Step 9.5.1

Multiply $2$ by $13$ .

$\frac{12 + 26}{13}$

Step 9.5.2

Add $12$ and $26$ .

$\frac{38}{13}$

Step 10

$x = 13$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 13$ is a local minimum

Step 11

Find the y-value when $x = 13$ .

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Step 11.1

Replace the variable $x$ with $13$ in the expression.

$f (13) = {(13)}^{2} - 20 \cdot 13 + 110 + \frac{1014}{13}$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify each term.

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Step 11.2.1.1

Raise $13$ to the power of $2$ .

$f (13) = 169 - 20 \cdot 13 + 110 + \frac{1014}{13}$

Step 11.2.1.2

Multiply $- 20$ by $13$ .

$f (13) = 169 - 260 + 110 + \frac{1014}{13}$

Step 11.2.1.3

Divide $1014$ by $13$ .

$f (13) = 169 - 260 + 110 + 78$

Step 11.2.2

Simplify by adding and subtracting.

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Step 11.2.2.1

Subtract $260$ from $169$ .

$f (13) = - 91 + 110 + 78$

Step 11.2.2.2

Add $- 91$ and $110$ .

$f (13) = 19 + 78$

Step 11.2.2.3

Add $19$ and $78$ .

$f (13) = 97$

Step 11.2.3

The final answer is $97$ .

$y = 97$

Step 12

These are the local extrema for $f (x) = x^{2} - 20 x + 110 + \frac{1014}{x}$ .

$(13, 97)$ is a local minima

Step 13