Calculus Examples

$f (x) = 5 + \frac{1}{9} x + \frac{10000}{x}$

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 $5 + \frac{1}{9} x + \frac{10000}{x}$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [\frac{1}{9} x] + \frac{d}{d x} [\frac{10000}{x}]$ .

$\frac{d}{d x} [5] + \frac{d}{d x} [\frac{1}{9} x] + \frac{d}{d x} [\frac{10000}{x}]$

Step 1.1.2

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [\frac{1}{9} x] + \frac{d}{d x} [\frac{10000}{x}]$

Step 1.2

Evaluate $\frac{d}{d x} [\frac{1}{9} x]$ .

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Step 1.2.1

Since $\frac{1}{9}$ is constant with respect to $x$ , the derivative of $\frac{1}{9} x$ with respect to $x$ is $\frac{1}{9} \frac{d}{d x} [x]$ .

$0 + \frac{1}{9} \frac{d}{d x} [x] + \frac{d}{d x} [\frac{10000}{x}]$

Step 1.2.2

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

$0 + \frac{1}{9} \cdot 1 + \frac{d}{d x} [\frac{10000}{x}]$

Step 1.2.3

Multiply $\frac{1}{9}$ by $1$ .

$0 + \frac{1}{9} + \frac{d}{d x} [\frac{10000}{x}]$

Step 1.3

Evaluate $\frac{d}{d x} [\frac{10000}{x}]$ .

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Step 1.3.1

Since $10000$ is constant with respect to $x$ , the derivative of $\frac{10000}{x}$ with respect to $x$ is $10000 \frac{d}{d x} [\frac{1}{x}]$ .

$0 + \frac{1}{9} + 10000 \frac{d}{d x} [\frac{1}{x}]$

Step 1.3.2

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

$0 + \frac{1}{9} + 10000 \frac{d}{d x} [x^{- 1}]$

Step 1.3.3

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

$0 + \frac{1}{9} + 10000 (- x^{- 2})$

Step 1.3.4

Multiply $- 1$ by $10000$ .

$0 + \frac{1}{9} - 10000 x^{- 2}$

Step 1.4

Simplify.

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Step 1.4.1

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

$0 + \frac{1}{9} - 10000 \frac{1}{x^{2}}$

Step 1.4.2

Combine terms.

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Step 1.4.2.1

Add $0$ and $\frac{1}{9}$ .

$\frac{1}{9} - 10000 \frac{1}{x^{2}}$

Step 1.4.2.2

Combine $- 10000$ and $\frac{1}{x^{2}}$ .

$\frac{1}{9} + \frac{- 10000}{x^{2}}$

Step 1.4.2.3

Move the negative in front of the fraction.

$\frac{1}{9} - \frac{10000}{x^{2}}$

Step 1.4.3

Reorder terms.

$- \frac{10000}{x^{2}} + \frac{1}{9}$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $- \frac{10000}{x^{2}} + \frac{1}{9}$ with respect to $x$ is $\frac{d}{d x} [- \frac{10000}{x^{2}}] + \frac{d}{d x} [\frac{1}{9}]$ .

$f'' (x) = \frac{d}{d x} (- \frac{10000}{x^{2}}) + \frac{d}{d x} (\frac{1}{9})$

Step 2.2

Evaluate $\frac{d}{d x} [- \frac{10000}{x^{2}}]$ .

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Step 2.2.1

Since $- 10000$ is constant with respect to $x$ , the derivative of $- \frac{10000}{x^{2}}$ with respect to $x$ is $- 10000 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

$f'' (x) = - 10000 \frac{d}{d x} \frac{1}{x^{2}} + \frac{d}{d x} (\frac{1}{9})$

Step 2.2.2

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

$f'' (x) = - 10000 \frac{d}{d x} {(x^{2})}^{- 1} + \frac{d}{d x} (\frac{1}{9})$

Step 2.2.3

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

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Step 2.2.3.1

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

$f'' (x) = - 10000 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{2})) + \frac{d}{d x} (\frac{1}{9})$

Step 2.2.3.2

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

$f'' (x) = - 10000 (- u^{- 2} \frac{d}{d x} x^{2}) + \frac{d}{d x} (\frac{1}{9})$

Step 2.2.3.3

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

$f'' (x) = - 10000 (- {(x^{2})}^{- 2} \frac{d}{d x} x^{2}) + \frac{d}{d x} (\frac{1}{9})$

Step 2.2.4

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

$f'' (x) = - 10000 (- {(x^{2})}^{- 2} (2 x)) + \frac{d}{d x} (\frac{1}{9})$

Step 2.2.5

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

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Step 2.2.5.1

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

$f'' (x) = - 10000 (- x^{2 \cdot - 2} (2 x)) + \frac{d}{d x} (\frac{1}{9})$

Step 2.2.5.2

Multiply $2$ by $- 2$ .

$f'' (x) = - 10000 (- x^{- 4} (2 x)) + \frac{d}{d x} (\frac{1}{9})$

Step 2.2.6

Multiply $2$ by $- 1$ .

$f'' (x) = - 10000 (- 2 x^{- 4} x) + \frac{d}{d x} (\frac{1}{9})$

Step 2.2.7

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

$f'' (x) = - 10000 (- 2 (x \cdot x^{- 4})) + \frac{d}{d x} (\frac{1}{9})$

Step 2.2.8

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

$f'' (x) = - 10000 (- 2 x^{1 - 4}) + \frac{d}{d x} (\frac{1}{9})$

Step 2.2.9

Subtract $4$ from $1$ .

$f'' (x) = - 10000 (- 2 x^{- 3}) + \frac{d}{d x} (\frac{1}{9})$

Step 2.2.10

Multiply $- 2$ by $- 10000$ .

$f'' (x) = 20000 x^{- 3} + \frac{d}{d x} (\frac{1}{9})$

Step 2.3

Since $\frac{1}{9}$ is constant with respect to $x$ , the derivative of $\frac{1}{9}$ with respect to $x$ is $0$ .

$f'' (x) = 20000 x^{- 3} + 0$

Step 2.4

Simplify.

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Step 2.4.1

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

$f'' (x) = 20000 (\frac{1}{x^{3}}) + 0$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Combine $20000$ and $\frac{1}{x^{3}}$ .

$f'' (x) = \frac{20000}{x^{3}} + 0$

Step 2.4.2.2

Add $\frac{20000}{x^{3}}$ and $0$ .

$f'' (x) = \frac{20000}{x^{3}}$

Step 3

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

$- \frac{10000}{x^{2}} + \frac{1}{9} = 0$

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

By the Sum Rule, the derivative of $5 + \frac{1}{9} x + \frac{10000}{x}$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [\frac{1}{9} x] + \frac{d}{d x} [\frac{10000}{x}]$ .

$\frac{d}{d x} [5] + \frac{d}{d x} [\frac{1}{9} x] + \frac{d}{d x} [\frac{10000}{x}]$

Step 4.1.1.2

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [\frac{1}{9} x] + \frac{d}{d x} [\frac{10000}{x}]$

Step 4.1.2

Evaluate $\frac{d}{d x} [\frac{1}{9} x]$ .

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Step 4.1.2.1

Since $\frac{1}{9}$ is constant with respect to $x$ , the derivative of $\frac{1}{9} x$ with respect to $x$ is $\frac{1}{9} \frac{d}{d x} [x]$ .

$0 + \frac{1}{9} \frac{d}{d x} [x] + \frac{d}{d x} [\frac{10000}{x}]$

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 $n = 1$ .

$0 + \frac{1}{9} \cdot 1 + \frac{d}{d x} [\frac{10000}{x}]$

Step 4.1.2.3

Multiply $\frac{1}{9}$ by $1$ .

$0 + \frac{1}{9} + \frac{d}{d x} [\frac{10000}{x}]$

Step 4.1.3

Evaluate $\frac{d}{d x} [\frac{10000}{x}]$ .

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Step 4.1.3.1

Since $10000$ is constant with respect to $x$ , the derivative of $\frac{10000}{x}$ with respect to $x$ is $10000 \frac{d}{d x} [\frac{1}{x}]$ .

$0 + \frac{1}{9} + 10000 \frac{d}{d x} [\frac{1}{x}]$

Step 4.1.3.2

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

$0 + \frac{1}{9} + 10000 \frac{d}{d x} [x^{- 1}]$

Step 4.1.3.3

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

$0 + \frac{1}{9} + 10000 (- x^{- 2})$

Step 4.1.3.4

Multiply $- 1$ by $10000$ .

$0 + \frac{1}{9} - 10000 x^{- 2}$

Step 4.1.4

Simplify.

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Step 4.1.4.1

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

$0 + \frac{1}{9} - 10000 \frac{1}{x^{2}}$

Step 4.1.4.2

Combine terms.

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Step 4.1.4.2.1

Add $0$ and $\frac{1}{9}$ .

$\frac{1}{9} - 10000 \frac{1}{x^{2}}$

Step 4.1.4.2.2

Combine $- 10000$ and $\frac{1}{x^{2}}$ .

$\frac{1}{9} + \frac{- 10000}{x^{2}}$

Step 4.1.4.2.3

Move the negative in front of the fraction.

$\frac{1}{9} - \frac{10000}{x^{2}}$

Step 4.1.4.3

Reorder terms.

$f' (x) = - \frac{10000}{x^{2}} + \frac{1}{9}$

Step 4.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{10000}{x^{2}} + \frac{1}{9}$ .

$- \frac{10000}{x^{2}} + \frac{1}{9}$

Step 5

Set the first derivative equal to $0$ then solve the equation $- \frac{10000}{x^{2}} + \frac{1}{9} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$- \frac{10000}{x^{2}} + \frac{1}{9} = 0$

Step 5.2

Subtract $\frac{1}{9}$ from both sides of the equation.

$- \frac{10000}{x^{2}} = - \frac{1}{9}$

Step 5.3

Find the LCD of the terms in the equation.

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Step 5.3.1

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

$x^{2}, 9$

Step 5.3.2

Since $x^{2}, 9$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 9$ then find LCM for the variable part $x^{2}$ .

Step 5.3.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 5.3.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 5.3.5

$9$ has factors of $3$ and $3$ .

$3 \cdot 3$

Step 5.3.6

Multiply $3$ by $3$ .

$9$

Step 5.3.7

The factors for $x^{2}$ are $x \cdot x$ , which is $x$ multiplied by each other $2$ times.

$x^{2} = x \cdot x$

$x$ occurs $2$ times.

Step 5.3.8

The LCM of $x^{2}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x \cdot x$

Step 5.3.9

Multiply $x$ by $x$ .

$x^{2}$

Step 5.3.10

The LCM for $x^{2}, 9$ is the numeric part $9$ multiplied by the variable part.

$9 x^{2}$

Step 5.4

Multiply each term in $- \frac{10000}{x^{2}} = - \frac{1}{9}$ by $9 x^{2}$ to eliminate the fractions.

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Step 5.4.1

Multiply each term in $- \frac{10000}{x^{2}} = - \frac{1}{9}$ by $9 x^{2}$ .

$- \frac{10000}{x^{2}} (9 x^{2}) = - \frac{1}{9} (9 x^{2})$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

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

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Step 5.4.2.1.1

Move the leading negative in $- \frac{10000}{x^{2}}$ into the numerator.

$\frac{- 10000}{x^{2}} (9 x^{2}) = - \frac{1}{9} (9 x^{2})$

Step 5.4.2.1.2

Factor $x^{2}$ out of $9 x^{2}$ .

$\frac{- 10000}{x^{2}} (x^{2} \cdot 9) = - \frac{1}{9} (9 x^{2})$

Step 5.4.2.1.3

Cancel the common factor.

$\frac{- 10000}{x^{2}} (x^{2} \cdot 9) = - \frac{1}{9} (9 x^{2})$

Step 5.4.2.1.4

Rewrite the expression.

$- 10000 \cdot 9 = - \frac{1}{9} (9 x^{2})$

Step 5.4.2.2

Multiply $- 10000$ by $9$ .

$- 90000 = - \frac{1}{9} (9 x^{2})$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Cancel the common factor of $9$ .

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Step 5.4.3.1.1

Move the leading negative in $- \frac{1}{9}$ into the numerator.

$- 90000 = \frac{- 1}{9} (9 x^{2})$

Step 5.4.3.1.2

Factor $9$ out of $9 x^{2}$ .

$- 90000 = \frac{- 1}{9} (9 (x^{2}))$

Step 5.4.3.1.3

Cancel the common factor.

$- 90000 = \frac{- 1}{9} (9 x^{2})$

Step 5.4.3.1.4

Rewrite the expression.

$- 90000 = - x^{2}$

Step 5.5

Solve the equation.

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Step 5.5.1

Rewrite the equation as $- x^{2} = - 90000$ .

$- x^{2} = - 90000$

Step 5.5.2

Divide each term in $- x^{2} = - 90000$ by $- 1$ and simplify.

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Step 5.5.2.1

Divide each term in $- x^{2} = - 90000$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 90000}{- 1}$

Step 5.5.2.2

Simplify the left side.

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Step 5.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 90000}{- 1}$

Step 5.5.2.2.2

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

$x^{2} = \frac{- 90000}{- 1}$

Step 5.5.2.3

Simplify the right side.

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Step 5.5.2.3.1

Divide $- 90000$ by $- 1$ .

$x^{2} = 90000$

Step 5.5.3

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

$x = \pm \sqrt{90000}$

Step 5.5.4

Simplify $\pm \sqrt{90000}$ .

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Step 5.5.4.1

Rewrite $90000$ as $300^{2}$ .

$x = \pm \sqrt{300^{2}}$

Step 5.5.4.2

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

$x = \pm 300$

Step 5.5.5

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

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Step 5.5.5.1

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

$x = 300$

Step 5.5.5.2

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

$x = - 300$

Step 5.5.5.3

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

$x = 300, - 300$

Step 6

Find the values where the derivative is undefined.

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Step 6.1

Set the denominator in $\frac{10000}{x^{2}}$ 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.

$x = 300, - 300$

Step 8

Evaluate the second derivative at $x = 300$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{20000}{{(300)}^{3}}$

Step 9

Evaluate the second derivative.

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Step 9.1

Raise $300$ to the power of $3$ .

$\frac{20000}{27000000}$

Step 9.2

Cancel the common factor of $20000$ and $27000000$ .

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Step 9.2.1

Factor $20000$ out of $20000$ .

$\frac{20000 (1)}{27000000}$

Step 9.2.2

Cancel the common factors.

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Step 9.2.2.1

Factor $20000$ out of $27000000$ .

$\frac{20000 \cdot 1}{20000 \cdot 1350}$

Step 9.2.2.2

Cancel the common factor.

$\frac{20000 \cdot 1}{20000 \cdot 1350}$

Step 9.2.2.3

Rewrite the expression.

$\frac{1}{1350}$

Step 10

$x = 300$ is a local minimum because the value of the second derivative is positive. This is referred to as the second derivative test.

$x = 300$ is a local minimum

Step 11

Find the y-value when $x = 300$ .

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Step 11.1

Replace the variable $x$ with $300$ in the expression.

$f (300) = 5 + \frac{1}{9} \cdot (300) + \frac{10000}{300}$

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

Cancel the common factor of $3$ .

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Step 11.2.1.1.1

Factor $3$ out of $9$ .

$f (300) = 5 + \frac{1}{3 (3)} \cdot 300 + \frac{10000}{300}$

Step 11.2.1.1.2

Factor $3$ out of $300$ .

$f (300) = 5 + \frac{1}{3 \cdot 3} \cdot (3 \cdot 100) + \frac{10000}{300}$

Step 11.2.1.1.3

Cancel the common factor.

$f (300) = 5 + \frac{1}{3 \cdot 3} \cdot (3 \cdot 100) + \frac{10000}{300}$

Step 11.2.1.1.4

Rewrite the expression.

$f (300) = 5 + \frac{1}{3} \cdot 100 + \frac{10000}{300}$

Step 11.2.1.2

Combine $\frac{1}{3}$ and $100$ .

$f (300) = 5 + \frac{100}{3} + \frac{10000}{300}$

Step 11.2.1.3

Cancel the common factor of $10000$ and $300$ .

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Step 11.2.1.3.1

Factor $100$ out of $10000$ .

$f (300) = 5 + \frac{100}{3} + \frac{100 (100)}{300}$

Step 11.2.1.3.2

Cancel the common factors.

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Step 11.2.1.3.2.1

Factor $100$ out of $300$ .

$f (300) = 5 + \frac{100}{3} + \frac{100 \cdot 100}{100 \cdot 3}$

Step 11.2.1.3.2.2

Cancel the common factor.

$f (300) = 5 + \frac{100}{3} + \frac{100 \cdot 100}{100 \cdot 3}$

Step 11.2.1.3.2.3

Rewrite the expression.

$f (300) = 5 + \frac{100}{3} + \frac{100}{3}$

Step 11.2.2

Combine fractions.

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Step 11.2.2.1

Combine the numerators over the common denominator.

$f (300) = 5 + \frac{100 + 100}{3}$

Step 11.2.2.2

Add $100$ and $100$ .

$f (300) = 5 + \frac{200}{3}$

Step 11.2.3

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (300) = 5 \cdot \frac{3}{3} + \frac{200}{3}$

Step 11.2.4

Combine $5$ and $\frac{3}{3}$ .

$f (300) = \frac{5 \cdot 3}{3} + \frac{200}{3}$

Step 11.2.5

Combine the numerators over the common denominator.

$f (300) = \frac{5 \cdot 3 + 200}{3}$

Step 11.2.6

Simplify the numerator.

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Step 11.2.6.1

Multiply $5$ by $3$ .

$f (300) = \frac{15 + 200}{3}$

Step 11.2.6.2

Add $15$ and $200$ .

$f (300) = \frac{215}{3}$

Step 11.2.7

The final answer is $\frac{215}{3}$ .

$y = \frac{215}{3}$

Step 12

Evaluate the second derivative at $x = - 300$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$\frac{20000}{{(- 300)}^{3}}$

Step 13

Evaluate the second derivative.

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Step 13.1

Raise $- 300$ to the power of $3$ .

$\frac{20000}{- 27000000}$

Step 13.2

Cancel the common factor of $20000$ and $- 27000000$ .

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Step 13.2.1

Factor $20000$ out of $20000$ .

$\frac{20000 (1)}{- 27000000}$

Step 13.2.2

Cancel the common factors.

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Step 13.2.2.1

Factor $20000$ out of $- 27000000$ .

$\frac{20000 \cdot 1}{20000 \cdot - 1350}$

Step 13.2.2.2

Cancel the common factor.

$\frac{20000 \cdot 1}{20000 \cdot - 1350}$

Step 13.2.2.3

Rewrite the expression.

$\frac{1}{- 1350}$

Step 13.3

Move the negative in front of the fraction.

$- \frac{1}{1350}$

Step 14

$x = - 300$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = - 300$ is a local maximum

Step 15

Find the y-value when $x = - 300$ .

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Step 15.1

Replace the variable $x$ with $- 300$ in the expression.

$f (- 300) = 5 + \frac{1}{9} \cdot (- 300) + \frac{10000}{- 300}$

Step 15.2

Simplify the result.

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Step 15.2.1

Simplify each term.

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Step 15.2.1.1

Cancel the common factor of $3$ .

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Step 15.2.1.1.1

Factor $3$ out of $9$ .

$f (- 300) = 5 + \frac{1}{3 (3)} \cdot - 300 + \frac{10000}{- 300}$

Step 15.2.1.1.2

Factor $3$ out of $- 300$ .

$f (- 300) = 5 + \frac{1}{3 \cdot 3} \cdot (3 \cdot - 100) + \frac{10000}{- 300}$

Step 15.2.1.1.3

Cancel the common factor.

$f (- 300) = 5 + \frac{1}{3 \cdot 3} \cdot (3 \cdot - 100) + \frac{10000}{- 300}$

Step 15.2.1.1.4

Rewrite the expression.

$f (- 300) = 5 + \frac{1}{3} \cdot - 100 + \frac{10000}{- 300}$

Step 15.2.1.2

Combine $\frac{1}{3}$ and $- 100$ .

$f (- 300) = 5 + \frac{- 100}{3} + \frac{10000}{- 300}$

Step 15.2.1.3

Move the negative in front of the fraction.

$f (- 300) = 5 - \frac{100}{3} + \frac{10000}{- 300}$

Step 15.2.1.4

Cancel the common factor of $10000$ and $- 300$ .

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Step 15.2.1.4.1

Factor $100$ out of $10000$ .

$f (- 300) = 5 - \frac{100}{3} + \frac{100 (100)}{- 300}$

Step 15.2.1.4.2

Cancel the common factors.

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Step 15.2.1.4.2.1

Factor $100$ out of $- 300$ .

$f (- 300) = 5 - \frac{100}{3} + \frac{100 \cdot 100}{100 \cdot - 3}$

Step 15.2.1.4.2.2

Cancel the common factor.

$f (- 300) = 5 - \frac{100}{3} + \frac{100 \cdot 100}{100 \cdot - 3}$

Step 15.2.1.4.2.3

Rewrite the expression.

$f (- 300) = 5 - \frac{100}{3} + \frac{100}{- 3}$

Step 15.2.1.5

Move the negative in front of the fraction.

$f (- 300) = 5 - \frac{100}{3} - \frac{100}{3}$

Step 15.2.2

Combine fractions.

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Step 15.2.2.1

Combine the numerators over the common denominator.

$f (- 300) = 5 + \frac{- 100 - 100}{3}$

Step 15.2.2.2

Simplify the expression.

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Step 15.2.2.2.1

Subtract $100$ from $- 100$ .

$f (- 300) = 5 + \frac{- 200}{3}$

Step 15.2.2.2.2

Move the negative in front of the fraction.

$f (- 300) = 5 - \frac{200}{3}$

Step 15.2.3

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$f (- 300) = 5 \cdot \frac{3}{3} - \frac{200}{3}$

Step 15.2.4

Combine $5$ and $\frac{3}{3}$ .

$f (- 300) = \frac{5 \cdot 3}{3} - \frac{200}{3}$

Step 15.2.5

Combine the numerators over the common denominator.

$f (- 300) = \frac{5 \cdot 3 - 200}{3}$

Step 15.2.6

Simplify the numerator.

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Step 15.2.6.1

Multiply $5$ by $3$ .

$f (- 300) = \frac{15 - 200}{3}$

Step 15.2.6.2

Subtract $200$ from $15$ .

$f (- 300) = \frac{- 185}{3}$

Step 15.2.7

Move the negative in front of the fraction.

$f (- 300) = - \frac{185}{3}$

Step 15.2.8

The final answer is $- \frac{185}{3}$ .

$y = - \frac{185}{3}$

Step 16

These are the local extrema for $f (x) = 5 + \frac{1}{9} x + \frac{10000}{x}$ .

$(300, \frac{215}{3})$ is a local minima

$(- 300, - \frac{185}{3})$ is a local maxima

Step 17