Calculus Examples

$R (x) = 4000 {(1 - \frac{x}{300})}^{2}$

Step 1

Find the first derivative of the function.

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Step 1.1

Rewrite ${(1 - \frac{x}{300})}^{2}$ as $(1 - \frac{x}{300}) (1 - \frac{x}{300})$ .

$\frac{d}{d x} [4000 ((1 - \frac{x}{300}) (1 - \frac{x}{300}))]$

Step 1.2

Expand $(1 - \frac{x}{300}) (1 - \frac{x}{300})$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\frac{d}{d x} [4000 (1 (1 - \frac{x}{300}) - \frac{x}{300} (1 - \frac{x}{300}))]$

Step 1.2.2

Apply the distributive property.

$\frac{d}{d x} [4000 (1 \cdot 1 + 1 (- \frac{x}{300}) - \frac{x}{300} (1 - \frac{x}{300}))]$

Step 1.2.3

Apply the distributive property.

$\frac{d}{d x} [4000 (1 \cdot 1 + 1 (- \frac{x}{300}) - \frac{x}{300} \cdot 1 - \frac{x}{300} (- \frac{x}{300}))]$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $1$ by $1$ .

$\frac{d}{d x} [4000 (1 + 1 (- \frac{x}{300}) - \frac{x}{300} \cdot 1 - \frac{x}{300} (- \frac{x}{300}))]$

Step 1.3.1.2

Multiply $- \frac{x}{300}$ by $1$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{300} - \frac{x}{300} \cdot 1 - \frac{x}{300} (- \frac{x}{300}))]$

Step 1.3.1.3

Multiply $- 1$ by $1$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{300} - \frac{x}{300} - \frac{x}{300} (- \frac{x}{300}))]$

Step 1.3.1.4

Multiply $- \frac{x}{300} (- \frac{x}{300})$ .

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Step 1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{300} - \frac{x}{300} + 1 \frac{x}{300} \frac{x}{300})]$

Step 1.3.1.4.2

Multiply $\frac{x}{300}$ by $1$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{300} - \frac{x}{300} + \frac{x}{300} \cdot \frac{x}{300})]$

Step 1.3.1.4.3

Multiply $\frac{x}{300}$ by $\frac{x}{300}$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{300} - \frac{x}{300} + \frac{x \cdot x}{300 \cdot 300})]$

Step 1.3.1.4.4

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

$\frac{d}{d x} [4000 (1 - \frac{x}{300} - \frac{x}{300} + \frac{x^{1} x}{300 \cdot 300})]$

Step 1.3.1.4.5

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

$\frac{d}{d x} [4000 (1 - \frac{x}{300} - \frac{x}{300} + \frac{x^{1} x^{1}}{300 \cdot 300})]$

Step 1.3.1.4.6

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

$\frac{d}{d x} [4000 (1 - \frac{x}{300} - \frac{x}{300} + \frac{x^{1 + 1}}{300 \cdot 300})]$

Step 1.3.1.4.7

Add $1$ and $1$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{300} - \frac{x}{300} + \frac{x^{2}}{300 \cdot 300})]$

Step 1.3.1.4.8

Multiply $300$ by $300$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{300} - \frac{x}{300} + \frac{x^{2}}{90000})]$

Step 1.3.2

Subtract $\frac{x}{300}$ from $- \frac{x}{300}$ .

$\frac{d}{d x} [4000 (1 - 2 \frac{x}{300} + \frac{x^{2}}{90000})]$

Step 1.4

Simplify each term.

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Step 1.4.1

Cancel the common factor of $2$ .

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Step 1.4.1.1

Factor $2$ out of $- 2$ .

$\frac{d}{d x} [4000 (1 + 2 (- 1) \frac{x}{300} + \frac{x^{2}}{90000})]$

Step 1.4.1.2

Factor $2$ out of $300$ .

$\frac{d}{d x} [4000 (1 + 2 \cdot - 1 \frac{x}{2 \cdot 150} + \frac{x^{2}}{90000})]$

Step 1.4.1.3

Cancel the common factor.

$\frac{d}{d x} [4000 (1 + 2 \cdot - 1 \frac{x}{2 \cdot 150} + \frac{x^{2}}{90000})]$

Step 1.4.1.4

Rewrite the expression.

$\frac{d}{d x} [4000 (1 - 1 \frac{x}{150} + \frac{x^{2}}{90000})]$

Step 1.4.2

Rewrite $- 1 \frac{x}{150}$ as $- \frac{x}{150}$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{150} + \frac{x^{2}}{90000})]$

Step 1.5

Since $4000$ is constant with respect to $x$ , the derivative of $4000 (1 - \frac{x}{150} + \frac{x^{2}}{90000})$ with respect to $x$ is $4000 \frac{d}{d x} [1 - \frac{x}{150} + \frac{x^{2}}{90000}]$ .

$4000 \frac{d}{d x} [1 - \frac{x}{150} + \frac{x^{2}}{90000}]$

Step 1.6

By the Sum Rule, the derivative of $1 - \frac{x}{150} + \frac{x^{2}}{90000}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{x}{150}] + \frac{d}{d x} [\frac{x^{2}}{90000}]$ .

$4000 (\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{x}{150}] + \frac{d}{d x} [\frac{x^{2}}{90000}])$

Step 1.7

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

$4000 (0 + \frac{d}{d x} [- \frac{x}{150}] + \frac{d}{d x} [\frac{x^{2}}{90000}])$

Step 1.8

Add $0$ and $\frac{d}{d x} [- \frac{x}{150}]$ .

$4000 (\frac{d}{d x} [- \frac{x}{150}] + \frac{d}{d x} [\frac{x^{2}}{90000}])$

Step 1.9

Since $- \frac{1}{150}$ is constant with respect to $x$ , the derivative of $- \frac{x}{150}$ with respect to $x$ is $- \frac{1}{150} \frac{d}{d x} [x]$ .

$4000 (- \frac{1}{150} \frac{d}{d x} [x] + \frac{d}{d x} [\frac{x^{2}}{90000}])$

Step 1.10

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

$4000 (- \frac{1}{150} \cdot 1 + \frac{d}{d x} [\frac{x^{2}}{90000}])$

Step 1.11

Multiply $- 1$ by $1$ .

$4000 (- \frac{1}{150} + \frac{d}{d x} [\frac{x^{2}}{90000}])$

Step 1.12

Since $\frac{1}{90000}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{90000}$ with respect to $x$ is $\frac{1}{90000} \frac{d}{d x} [x^{2}]$ .

$4000 (- \frac{1}{150} + \frac{1}{90000} \frac{d}{d x} [x^{2}])$

Step 1.13

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

$4000 (- \frac{1}{150} + \frac{1}{90000} (2 x))$

Step 1.14

Simplify terms.

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Step 1.14.1

Combine $2$ and $\frac{1}{90000}$ .

$4000 (- \frac{1}{150} + \frac{2}{90000} x)$

Step 1.14.2

Combine $\frac{2}{90000}$ and $x$ .

$4000 (- \frac{1}{150} + \frac{2 x}{90000})$

Step 1.14.3

Cancel the common factor of $2$ and $90000$ .

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Step 1.14.3.1

Factor $2$ out of $2 x$ .

$4000 (- \frac{1}{150} + \frac{2 (x)}{90000})$

Step 1.14.3.2

Cancel the common factors.

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Step 1.14.3.2.1

Factor $2$ out of $90000$ .

$4000 (- \frac{1}{150} + \frac{2 x}{2 \cdot 45000})$

Step 1.14.3.2.2

Cancel the common factor.

$4000 (- \frac{1}{150} + \frac{2 x}{2 \cdot 45000})$

Step 1.14.3.2.3

Rewrite the expression.

$4000 (- \frac{1}{150} + \frac{x}{45000})$

Step 1.15

Simplify.

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Step 1.15.1

Apply the distributive property.

$4000 (- \frac{1}{150}) + 4000 \frac{x}{45000}$

Step 1.15.2

Combine terms.

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Step 1.15.2.1

Multiply $- 1$ by $4000$ .

$- 4000 (\frac{1}{150}) + 4000 \frac{x}{45000}$

Step 1.15.2.2

Combine $- 4000$ and $\frac{1}{150}$ .

$\frac{- 4000}{150} + 4000 \frac{x}{45000}$

Step 1.15.2.3

Cancel the common factor of $- 4000$ and $150$ .

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Step 1.15.2.3.1

Factor $50$ out of $- 4000$ .

$\frac{50 (- 80)}{150} + 4000 \frac{x}{45000}$

Step 1.15.2.3.2

Cancel the common factors.

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Step 1.15.2.3.2.1

Factor $50$ out of $150$ .

$\frac{50 \cdot - 80}{50 \cdot 3} + 4000 \frac{x}{45000}$

Step 1.15.2.3.2.2

Cancel the common factor.

$\frac{50 \cdot - 80}{50 \cdot 3} + 4000 \frac{x}{45000}$

Step 1.15.2.3.2.3

Rewrite the expression.

$\frac{- 80}{3} + 4000 \frac{x}{45000}$

Step 1.15.2.4

Move the negative in front of the fraction.

$- \frac{80}{3} + 4000 \frac{x}{45000}$

Step 1.15.2.5

Combine $4000$ and $\frac{x}{45000}$ .

$- \frac{80}{3} + \frac{4000 x}{45000}$

Step 1.15.2.6

Cancel the common factor of $4000$ and $45000$ .

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Step 1.15.2.6.1

Factor $1000$ out of $4000 x$ .

$- \frac{80}{3} + \frac{1000 (4 x)}{45000}$

Step 1.15.2.6.2

Cancel the common factors.

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Step 1.15.2.6.2.1

Factor $1000$ out of $45000$ .

$- \frac{80}{3} + \frac{1000 (4 x)}{1000 (45)}$

Step 1.15.2.6.2.2

Cancel the common factor.

$- \frac{80}{3} + \frac{1000 (4 x)}{1000 \cdot 45}$

Step 1.15.2.6.2.3

Rewrite the expression.

$- \frac{80}{3} + \frac{4 x}{45}$

Step 1.15.3

Reorder terms.

$\frac{4 x}{45} - \frac{80}{3}$

Step 2

Find the second derivative of the function.

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Step 2.1

By the Sum Rule, the derivative of $\frac{4 x}{45} - \frac{80}{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{4 x}{45}] + \frac{d}{d x} [- \frac{80}{3}]$ .

$R'' (x) = \frac{d}{d x} (\frac{4 x}{45}) + \frac{d}{d x} (- \frac{80}{3})$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{4 x}{45}]$ .

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Step 2.2.1

Since $\frac{4}{45}$ is constant with respect to $x$ , the derivative of $\frac{4 x}{45}$ with respect to $x$ is $\frac{4}{45} \frac{d}{d x} [x]$ .

$R'' (x) = \frac{4}{45} \cdot \frac{d}{d x} (x) + \frac{d}{d x} (- \frac{80}{3})$

Step 2.2.2

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

$R'' (x) = \frac{4}{45} \cdot 1 + \frac{d}{d x} (- \frac{80}{3})$

Step 2.2.3

Multiply $\frac{4}{45}$ by $1$ .

$R'' (x) = \frac{4}{45} + \frac{d}{d x} (- \frac{80}{3})$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $- \frac{80}{3}$ is constant with respect to $x$ , the derivative of $- \frac{80}{3}$ with respect to $x$ is $0$ .

$R'' (x) = \frac{4}{45} + 0$

Step 2.3.2

Add $\frac{4}{45}$ and $0$ .

$R'' (x) = \frac{4}{45}$

Step 3

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

$\frac{4 x}{45} - \frac{80}{3} = 0$

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

Rewrite ${(1 - \frac{x}{300})}^{2}$ as $(1 - \frac{x}{300}) (1 - \frac{x}{300})$ .

$\frac{d}{d x} [4000 ((1 - \frac{x}{300}) (1 - \frac{x}{300}))]$

Step 4.1.2

Expand $(1 - \frac{x}{300}) (1 - \frac{x}{300})$ using the FOIL Method.

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Step 4.1.2.1

Apply the distributive property.

$\frac{d}{d x} [4000 (1 (1 - \frac{x}{300}) - \frac{x}{300} (1 - \frac{x}{300}))]$

Step 4.1.2.2

Apply the distributive property.

$\frac{d}{d x} [4000 (1 \cdot 1 + 1 (- \frac{x}{300}) - \frac{x}{300} (1 - \frac{x}{300}))]$

Step 4.1.2.3

Apply the distributive property.

$\frac{d}{d x} [4000 (1 \cdot 1 + 1 (- \frac{x}{300}) - \frac{x}{300} \cdot 1 - \frac{x}{300} (- \frac{x}{300}))]$

Step 4.1.3

Simplify and combine like terms.

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Step 4.1.3.1

Simplify each term.

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Step 4.1.3.1.1

Multiply $1$ by $1$ .

$\frac{d}{d x} [4000 (1 + 1 (- \frac{x}{300}) - \frac{x}{300} \cdot 1 - \frac{x}{300} (- \frac{x}{300}))]$

Step 4.1.3.1.2

Multiply $- \frac{x}{300}$ by $1$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{300} - \frac{x}{300} \cdot 1 - \frac{x}{300} (- \frac{x}{300}))]$

Step 4.1.3.1.3

Multiply $- 1$ by $1$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{300} - \frac{x}{300} - \frac{x}{300} (- \frac{x}{300}))]$

Step 4.1.3.1.4

Multiply $- \frac{x}{300} (- \frac{x}{300})$ .

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Step 4.1.3.1.4.1

Multiply $- 1$ by $- 1$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{300} - \frac{x}{300} + 1 \frac{x}{300} \frac{x}{300})]$

Step 4.1.3.1.4.2

Multiply $\frac{x}{300}$ by $1$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{300} - \frac{x}{300} + \frac{x}{300} \cdot \frac{x}{300})]$

Step 4.1.3.1.4.3

Multiply $\frac{x}{300}$ by $\frac{x}{300}$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{300} - \frac{x}{300} + \frac{x \cdot x}{300 \cdot 300})]$

Step 4.1.3.1.4.4

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

$\frac{d}{d x} [4000 (1 - \frac{x}{300} - \frac{x}{300} + \frac{x^{1} x}{300 \cdot 300})]$

Step 4.1.3.1.4.5

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

$\frac{d}{d x} [4000 (1 - \frac{x}{300} - \frac{x}{300} + \frac{x^{1} x^{1}}{300 \cdot 300})]$

Step 4.1.3.1.4.6

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

$\frac{d}{d x} [4000 (1 - \frac{x}{300} - \frac{x}{300} + \frac{x^{1 + 1}}{300 \cdot 300})]$

Step 4.1.3.1.4.7

Add $1$ and $1$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{300} - \frac{x}{300} + \frac{x^{2}}{300 \cdot 300})]$

Step 4.1.3.1.4.8

Multiply $300$ by $300$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{300} - \frac{x}{300} + \frac{x^{2}}{90000})]$

Step 4.1.3.2

Subtract $\frac{x}{300}$ from $- \frac{x}{300}$ .

$\frac{d}{d x} [4000 (1 - 2 \frac{x}{300} + \frac{x^{2}}{90000})]$

Step 4.1.4

Simplify each term.

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Step 4.1.4.1

Cancel the common factor of $2$ .

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Step 4.1.4.1.1

Factor $2$ out of $- 2$ .

$\frac{d}{d x} [4000 (1 + 2 (- 1) \frac{x}{300} + \frac{x^{2}}{90000})]$

Step 4.1.4.1.2

Factor $2$ out of $300$ .

$\frac{d}{d x} [4000 (1 + 2 \cdot - 1 \frac{x}{2 \cdot 150} + \frac{x^{2}}{90000})]$

Step 4.1.4.1.3

Cancel the common factor.

$\frac{d}{d x} [4000 (1 + 2 \cdot - 1 \frac{x}{2 \cdot 150} + \frac{x^{2}}{90000})]$

Step 4.1.4.1.4

Rewrite the expression.

$\frac{d}{d x} [4000 (1 - 1 \frac{x}{150} + \frac{x^{2}}{90000})]$

Step 4.1.4.2

Rewrite $- 1 \frac{x}{150}$ as $- \frac{x}{150}$ .

$\frac{d}{d x} [4000 (1 - \frac{x}{150} + \frac{x^{2}}{90000})]$

Step 4.1.5

$4000 \frac{d}{d x} [1 - \frac{x}{150} + \frac{x^{2}}{90000}]$

Step 4.1.6

By the Sum Rule, the derivative of $1 - \frac{x}{150} + \frac{x^{2}}{90000}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{x}{150}] + \frac{d}{d x} [\frac{x^{2}}{90000}]$ .

$4000 (\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{x}{150}] + \frac{d}{d x} [\frac{x^{2}}{90000}])$

Step 4.1.7

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

$4000 (0 + \frac{d}{d x} [- \frac{x}{150}] + \frac{d}{d x} [\frac{x^{2}}{90000}])$

Step 4.1.8

Add $0$ and $\frac{d}{d x} [- \frac{x}{150}]$ .

$4000 (\frac{d}{d x} [- \frac{x}{150}] + \frac{d}{d x} [\frac{x^{2}}{90000}])$

Step 4.1.9

Since $- \frac{1}{150}$ is constant with respect to $x$ , the derivative of $- \frac{x}{150}$ with respect to $x$ is $- \frac{1}{150} \frac{d}{d x} [x]$ .

$4000 (- \frac{1}{150} \frac{d}{d x} [x] + \frac{d}{d x} [\frac{x^{2}}{90000}])$

Step 4.1.10

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

$4000 (- \frac{1}{150} \cdot 1 + \frac{d}{d x} [\frac{x^{2}}{90000}])$

Step 4.1.11

Multiply $- 1$ by $1$ .

$4000 (- \frac{1}{150} + \frac{d}{d x} [\frac{x^{2}}{90000}])$

Step 4.1.12

Since $\frac{1}{90000}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{90000}$ with respect to $x$ is $\frac{1}{90000} \frac{d}{d x} [x^{2}]$ .

$4000 (- \frac{1}{150} + \frac{1}{90000} \frac{d}{d x} [x^{2}])$

Step 4.1.13

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

$4000 (- \frac{1}{150} + \frac{1}{90000} (2 x))$

Step 4.1.14

Simplify terms.

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Step 4.1.14.1

Combine $2$ and $\frac{1}{90000}$ .

$4000 (- \frac{1}{150} + \frac{2}{90000} x)$

Step 4.1.14.2

Combine $\frac{2}{90000}$ and $x$ .

$4000 (- \frac{1}{150} + \frac{2 x}{90000})$

Step 4.1.14.3

Cancel the common factor of $2$ and $90000$ .

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Step 4.1.14.3.1

Factor $2$ out of $2 x$ .

$4000 (- \frac{1}{150} + \frac{2 (x)}{90000})$

Step 4.1.14.3.2

Cancel the common factors.

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Step 4.1.14.3.2.1

Factor $2$ out of $90000$ .

$4000 (- \frac{1}{150} + \frac{2 x}{2 \cdot 45000})$

Step 4.1.14.3.2.2

Cancel the common factor.

$4000 (- \frac{1}{150} + \frac{2 x}{2 \cdot 45000})$

Step 4.1.14.3.2.3

Rewrite the expression.

$4000 (- \frac{1}{150} + \frac{x}{45000})$

Step 4.1.15

Simplify.

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Step 4.1.15.1

Apply the distributive property.

$4000 (- \frac{1}{150}) + 4000 \frac{x}{45000}$

Step 4.1.15.2

Combine terms.

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Step 4.1.15.2.1

Multiply $- 1$ by $4000$ .

$- 4000 (\frac{1}{150}) + 4000 \frac{x}{45000}$

Step 4.1.15.2.2

Combine $- 4000$ and $\frac{1}{150}$ .

$\frac{- 4000}{150} + 4000 \frac{x}{45000}$

Step 4.1.15.2.3

Cancel the common factor of $- 4000$ and $150$ .

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Step 4.1.15.2.3.1

Factor $50$ out of $- 4000$ .

$\frac{50 (- 80)}{150} + 4000 \frac{x}{45000}$

Step 4.1.15.2.3.2

Cancel the common factors.

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Step 4.1.15.2.3.2.1

Factor $50$ out of $150$ .

$\frac{50 \cdot - 80}{50 \cdot 3} + 4000 \frac{x}{45000}$

Step 4.1.15.2.3.2.2

Cancel the common factor.

$\frac{50 \cdot - 80}{50 \cdot 3} + 4000 \frac{x}{45000}$

Step 4.1.15.2.3.2.3

Rewrite the expression.

$\frac{- 80}{3} + 4000 \frac{x}{45000}$

Step 4.1.15.2.4

Move the negative in front of the fraction.

$- \frac{80}{3} + 4000 \frac{x}{45000}$

Step 4.1.15.2.5

Combine $4000$ and $\frac{x}{45000}$ .

$- \frac{80}{3} + \frac{4000 x}{45000}$

Step 4.1.15.2.6

Cancel the common factor of $4000$ and $45000$ .

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Step 4.1.15.2.6.1

Factor $1000$ out of $4000 x$ .

$- \frac{80}{3} + \frac{1000 (4 x)}{45000}$

Step 4.1.15.2.6.2

Cancel the common factors.

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Step 4.1.15.2.6.2.1

Factor $1000$ out of $45000$ .

$- \frac{80}{3} + \frac{1000 (4 x)}{1000 (45)}$

Step 4.1.15.2.6.2.2

Cancel the common factor.

$- \frac{80}{3} + \frac{1000 (4 x)}{1000 \cdot 45}$

Step 4.1.15.2.6.2.3

Rewrite the expression.

$- \frac{80}{3} + \frac{4 x}{45}$

Step 4.1.15.3

Reorder terms.

$f' (x) = \frac{4 x}{45} - \frac{80}{3}$

Step 4.2

The first derivative of $R (x)$ with respect to $x$ is $\frac{4 x}{45} - \frac{80}{3}$ .

$\frac{4 x}{45} - \frac{80}{3}$

Step 5

Set the first derivative equal to $0$ then solve the equation $\frac{4 x}{45} - \frac{80}{3} = 0$ .

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Step 5.1

Set the first derivative equal to $0$ .

$\frac{4 x}{45} - \frac{80}{3} = 0$

Step 5.2

Add $\frac{80}{3}$ to both sides of the equation.

$\frac{4 x}{45} = \frac{80}{3}$

Step 5.3

Multiply both sides of the equation by $\frac{45}{4}$ .

$\frac{45}{4} \cdot \frac{4 x}{45} = \frac{45}{4} \cdot \frac{80}{3}$

Step 5.4

Simplify both sides of the equation.

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Step 5.4.1

Simplify the left side.

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Step 5.4.1.1

Simplify $\frac{45}{4} \cdot \frac{4 x}{45}$ .

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Step 5.4.1.1.1

Cancel the common factor of $45$ .

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Step 5.4.1.1.1.1

Cancel the common factor.

$\frac{45}{4} \cdot \frac{4 x}{45} = \frac{45}{4} \cdot \frac{80}{3}$

Step 5.4.1.1.1.2

Rewrite the expression.

$\frac{1}{4} (4 x) = \frac{45}{4} \cdot \frac{80}{3}$

Step 5.4.1.1.2

Cancel the common factor of $4$ .

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Step 5.4.1.1.2.1

Factor $4$ out of $4 x$ .

$\frac{1}{4} (4 (x)) = \frac{45}{4} \cdot \frac{80}{3}$

Step 5.4.1.1.2.2

Cancel the common factor.

$\frac{1}{4} (4 x) = \frac{45}{4} \cdot \frac{80}{3}$

Step 5.4.1.1.2.3

Rewrite the expression.

$x = \frac{45}{4} \cdot \frac{80}{3}$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Simplify $\frac{45}{4} \cdot \frac{80}{3}$ .

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Step 5.4.2.1.1

Cancel the common factor of $3$ .

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Step 5.4.2.1.1.1

Factor $3$ out of $45$ .

$x = \frac{3 (15)}{4} \cdot \frac{80}{3}$

Step 5.4.2.1.1.2

Cancel the common factor.

$x = \frac{3 \cdot 15}{4} \cdot \frac{80}{3}$

Step 5.4.2.1.1.3

Rewrite the expression.

$x = \frac{15}{4} \cdot 80$

Step 5.4.2.1.2

Cancel the common factor of $4$ .

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Step 5.4.2.1.2.1

Factor $4$ out of $80$ .

$x = \frac{15}{4} \cdot (4 (20))$

Step 5.4.2.1.2.2

Cancel the common factor.

$x = \frac{15}{4} \cdot (4 \cdot 20)$

Step 5.4.2.1.2.3

Rewrite the expression.

$x = 15 \cdot 20$

Step 5.4.2.1.3

Multiply $15$ by $20$ .

$x = 300$

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.

$x = 300$

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.

$\frac{4}{45}$

Step 9

$x = 300$ 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 10

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

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Step 10.1

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

$R (300) = 4000 {(1 - \frac{300}{300})}^{2}$

Step 10.2

Simplify the result.

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Step 10.2.1

Simplify each term.

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Step 10.2.1.1

Cancel the common factor of $300$ .

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Step 10.2.1.1.1

Cancel the common factor.

$R (300) = 4000 {(1 - \frac{300}{300})}^{2}$

Step 10.2.1.1.2

Rewrite the expression.

$R (300) = 4000 {(1 - 1 \cdot 1)}^{2}$

Step 10.2.1.2

Multiply $- 1$ by $1$ .

$R (300) = 4000 {(1 - 1)}^{2}$

Step 10.2.2

Simplify the expression.

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Step 10.2.2.1

Subtract $1$ from $1$ .

$R (300) = 4000 \cdot 0^{2}$

Step 10.2.2.2

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

$R (300) = 4000 \cdot 0$

Step 10.2.2.3

Multiply $4000$ by $0$ .

$R (300) = 0$

Step 10.2.3

The final answer is $0$ .

$y = 0$

Step 11

These are the local extrema for $R (x) = 4000 {(1 - \frac{x}{300})}^{2}$ .

$(300, 0)$ is a local minima

Step 12