Calculus Examples

$y = 2 (700 - \frac{4 x}{3}) x$

Step 1

Write $y = 2 (700 - \frac{4 x}{3}) x$ as a function.

$f (x) = 2 (700 - \frac{4 x}{3}) x$

Step 2

Find the first derivative of the function.

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Step 2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 (700 - \frac{4 x}{3}) x$ with respect to $x$ is $2 \frac{d}{d x} [(700 - \frac{4 x}{3}) x]$ .

$2 \frac{d}{d x} [(700 - \frac{4 x}{3}) x]$

Step 2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = 700 - \frac{4 x}{3}$ and $g (x) = x$ .

$2 ((700 - \frac{4 x}{3}) \frac{d}{d x} [x] + x \frac{d}{d x} [700 - \frac{4 x}{3}])$

Step 2.3

Differentiate.

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Step 2.3.1

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

$2 ((700 - \frac{4 x}{3}) \cdot 1 + x \frac{d}{d x} [700 - \frac{4 x}{3}])$

Step 2.3.2

Multiply $700 - \frac{4 x}{3}$ by $1$ .

$2 (700 - \frac{4 x}{3} + x \frac{d}{d x} [700 - \frac{4 x}{3}])$

Step 2.3.3

By the Sum Rule, the derivative of $700 - \frac{4 x}{3}$ with respect to $x$ is $\frac{d}{d x} [700] + \frac{d}{d x} [- \frac{4 x}{3}]$ .

$2 (700 - \frac{4 x}{3} + x (\frac{d}{d x} [700] + \frac{d}{d x} [- \frac{4 x}{3}]))$

Step 2.3.4

Since $700$ is constant with respect to $x$ , the derivative of $700$ with respect to $x$ is $0$ .

$2 (700 - \frac{4 x}{3} + x (0 + \frac{d}{d x} [- \frac{4 x}{3}]))$

Step 2.3.5

Add $0$ and $\frac{d}{d x} [- \frac{4 x}{3}]$ .

$2 (700 - \frac{4 x}{3} + x \frac{d}{d x} [- \frac{4 x}{3}])$

Step 2.3.6

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

$2 (700 - \frac{4 x}{3} + x (- \frac{4}{3} \frac{d}{d x} [x]))$

Step 2.3.7

Combine fractions.

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Step 2.3.7.1

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

$2 (700 - \frac{4 x}{3} - \frac{x \cdot 4}{3} \frac{d}{d x} [x])$

Step 2.3.7.2

Move $4$ to the left of $x$ .

$2 (700 - \frac{4 x}{3} - \frac{4 \cdot x}{3} \frac{d}{d x} [x])$

Step 2.3.8

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

$2 (700 - \frac{4 x}{3} - \frac{4 x}{3} \cdot 1)$

Step 2.3.9

Simplify terms.

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Step 2.3.9.1

Multiply $- 1$ by $1$ .

$2 (700 - \frac{4 x}{3} - \frac{4 x}{3})$

Step 2.3.9.2

Subtract $\frac{4 x}{3}$ from $- \frac{4 x}{3}$ .

$2 (700 - 2 \frac{4 x}{3})$

Step 2.3.9.3

Combine $- 2$ and $\frac{4 x}{3}$ .

$2 (700 + \frac{- 2 (4 x)}{3})$

Step 2.3.9.4

Simplify the expression.

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Step 2.3.9.4.1

Multiply $4$ by $- 2$ .

$2 (700 + \frac{- 8 x}{3})$

Step 2.3.9.4.2

Move the negative in front of the fraction.

$2 (700 - \frac{8 x}{3})$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$2 \cdot 700 + 2 (- \frac{8 x}{3})$

Step 2.4.2

Combine terms.

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Step 2.4.2.1

Multiply $2$ by $700$ .

$1400 + 2 (- \frac{8 x}{3})$

Step 2.4.2.2

Multiply $- 1$ by $2$ .

$1400 - 2 \frac{8 x}{3}$

Step 2.4.2.3

Combine $- 2$ and $\frac{8 x}{3}$ .

$1400 + \frac{- 2 (8 x)}{3}$

Step 2.4.2.4

Multiply $8$ by $- 2$ .

$1400 + \frac{- 16 x}{3}$

Step 2.4.2.5

Move the negative in front of the fraction.

$1400 - \frac{16 x}{3}$

Step 2.4.3

Reorder terms.

$- \frac{16 x}{3} + 1400$

Step 3

Find the second derivative of the function.

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Step 3.1

By the Sum Rule, the derivative of $- \frac{16 x}{3} + 1400$ with respect to $x$ is $\frac{d}{d x} [- \frac{16 x}{3}] + \frac{d}{d x} [1400]$ .

$f'' (x) = \frac{d}{d x} (- \frac{16 x}{3}) + \frac{d}{d x} (1400)$

Step 3.2

Evaluate $\frac{d}{d x} [- \frac{16 x}{3}]$ .

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Step 3.2.1

Since $- \frac{16}{3}$ is constant with respect to $x$ , the derivative of $- \frac{16 x}{3}$ with respect to $x$ is $- \frac{16}{3} \frac{d}{d x} [x]$ .

$f'' (x) = - \frac{16}{3} \cdot \frac{d}{d x} x + \frac{d}{d x} (1400)$

Step 3.2.2

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

$f'' (x) = - \frac{16}{3} \cdot 1 + \frac{d}{d x} (1400)$

Step 3.2.3

Multiply $- 1$ by $1$ .

$f'' (x) = - \frac{16}{3} + \frac{d}{d x} (1400)$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

Since $1400$ is constant with respect to $x$ , the derivative of $1400$ with respect to $x$ is $0$ .

$f'' (x) = - \frac{16}{3} + 0$

Step 3.3.2

Add $- \frac{16}{3}$ and $0$ .

$f'' (x) = - \frac{16}{3}$

Step 4

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

$- \frac{16 x}{3} + 1400 = 0$

Step 5

Find the first derivative.

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Step 5.1

Find the first derivative.

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Step 5.1.1

Since $2$ is constant with respect to $x$ , the derivative of $2 (700 - \frac{4 x}{3}) x$ with respect to $x$ is $2 \frac{d}{d x} [(700 - \frac{4 x}{3}) x]$ .

$2 \frac{d}{d x} [(700 - \frac{4 x}{3}) x]$

Step 5.1.2

$2 ((700 - \frac{4 x}{3}) \frac{d}{d x} [x] + x \frac{d}{d x} [700 - \frac{4 x}{3}])$

Step 5.1.3

Differentiate.

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Step 5.1.3.1

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

$2 ((700 - \frac{4 x}{3}) \cdot 1 + x \frac{d}{d x} [700 - \frac{4 x}{3}])$

Step 5.1.3.2

Multiply $700 - \frac{4 x}{3}$ by $1$ .

$2 (700 - \frac{4 x}{3} + x \frac{d}{d x} [700 - \frac{4 x}{3}])$

Step 5.1.3.3

By the Sum Rule, the derivative of $700 - \frac{4 x}{3}$ with respect to $x$ is $\frac{d}{d x} [700] + \frac{d}{d x} [- \frac{4 x}{3}]$ .

$2 (700 - \frac{4 x}{3} + x (\frac{d}{d x} [700] + \frac{d}{d x} [- \frac{4 x}{3}]))$

Step 5.1.3.4

Since $700$ is constant with respect to $x$ , the derivative of $700$ with respect to $x$ is $0$ .

$2 (700 - \frac{4 x}{3} + x (0 + \frac{d}{d x} [- \frac{4 x}{3}]))$

Step 5.1.3.5

Add $0$ and $\frac{d}{d x} [- \frac{4 x}{3}]$ .

$2 (700 - \frac{4 x}{3} + x \frac{d}{d x} [- \frac{4 x}{3}])$

Step 5.1.3.6

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

$2 (700 - \frac{4 x}{3} + x (- \frac{4}{3} \frac{d}{d x} [x]))$

Step 5.1.3.7

Combine fractions.

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Step 5.1.3.7.1

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

$2 (700 - \frac{4 x}{3} - \frac{x \cdot 4}{3} \frac{d}{d x} [x])$

Step 5.1.3.7.2

Move $4$ to the left of $x$ .

$2 (700 - \frac{4 x}{3} - \frac{4 \cdot x}{3} \frac{d}{d x} [x])$

Step 5.1.3.8

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

$2 (700 - \frac{4 x}{3} - \frac{4 x}{3} \cdot 1)$

Step 5.1.3.9

Simplify terms.

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Step 5.1.3.9.1

Multiply $- 1$ by $1$ .

$2 (700 - \frac{4 x}{3} - \frac{4 x}{3})$

Step 5.1.3.9.2

Subtract $\frac{4 x}{3}$ from $- \frac{4 x}{3}$ .

$2 (700 - 2 \frac{4 x}{3})$

Step 5.1.3.9.3

Combine $- 2$ and $\frac{4 x}{3}$ .

$2 (700 + \frac{- 2 (4 x)}{3})$

Step 5.1.3.9.4

Simplify the expression.

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Step 5.1.3.9.4.1

Multiply $4$ by $- 2$ .

$2 (700 + \frac{- 8 x}{3})$

Step 5.1.3.9.4.2

Move the negative in front of the fraction.

$2 (700 - \frac{8 x}{3})$

Step 5.1.4

Simplify.

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Step 5.1.4.1

Apply the distributive property.

$2 \cdot 700 + 2 (- \frac{8 x}{3})$

Step 5.1.4.2

Combine terms.

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Step 5.1.4.2.1

Multiply $2$ by $700$ .

$1400 + 2 (- \frac{8 x}{3})$

Step 5.1.4.2.2

Multiply $- 1$ by $2$ .

$1400 - 2 \frac{8 x}{3}$

Step 5.1.4.2.3

Combine $- 2$ and $\frac{8 x}{3}$ .

$1400 + \frac{- 2 (8 x)}{3}$

Step 5.1.4.2.4

Multiply $8$ by $- 2$ .

$1400 + \frac{- 16 x}{3}$

Step 5.1.4.2.5

Move the negative in front of the fraction.

$1400 - \frac{16 x}{3}$

Step 5.1.4.3

Reorder terms.

$f' (x) = - \frac{16 x}{3} + 1400$

Step 5.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{16 x}{3} + 1400$ .

$- \frac{16 x}{3} + 1400$

Step 6

Set the first derivative equal to $0$ then solve the equation $- \frac{16 x}{3} + 1400 = 0$ .

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Step 6.1

Set the first derivative equal to $0$ .

$- \frac{16 x}{3} + 1400 = 0$

Step 6.2

Subtract $1400$ from both sides of the equation.

$- \frac{16 x}{3} = - 1400$

Step 6.3

Multiply both sides of the equation by $- \frac{3}{16}$ .

$- \frac{3}{16} (- \frac{16 x}{3}) = - \frac{3}{16} \cdot - 1400$

Step 6.4

Simplify both sides of the equation.

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Step 6.4.1

Simplify the left side.

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Step 6.4.1.1

Simplify $- \frac{3}{16} (- \frac{16 x}{3})$ .

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Step 6.4.1.1.1

Cancel the common factor of $3$ .

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Step 6.4.1.1.1.1

Move the leading negative in $- \frac{3}{16}$ into the numerator.

$\frac{- 3}{16} (- \frac{16 x}{3}) = - \frac{3}{16} \cdot - 1400$

Step 6.4.1.1.1.2

Move the leading negative in $- \frac{16 x}{3}$ into the numerator.

$\frac{- 3}{16} \cdot \frac{- 16 x}{3} = - \frac{3}{16} \cdot - 1400$

Step 6.4.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{16} \cdot \frac{- 16 x}{3} = - \frac{3}{16} \cdot - 1400$

Step 6.4.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{16} \cdot \frac{- 16 x}{3} = - \frac{3}{16} \cdot - 1400$

Step 6.4.1.1.1.5

Rewrite the expression.

$\frac{- 1}{16} (- 16 x) = - \frac{3}{16} \cdot - 1400$

Step 6.4.1.1.2

Cancel the common factor of $16$ .

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Step 6.4.1.1.2.1

Factor $16$ out of $- 16 x$ .

$\frac{- 1}{16} (16 (- x)) = - \frac{3}{16} \cdot - 1400$

Step 6.4.1.1.2.2

Cancel the common factor.

$\frac{- 1}{16} (16 (- x)) = - \frac{3}{16} \cdot - 1400$

Step 6.4.1.1.2.3

Rewrite the expression.

$- - x = - \frac{3}{16} \cdot - 1400$

Step 6.4.1.1.3

Multiply.

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Step 6.4.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 x = - \frac{3}{16} \cdot - 1400$

Step 6.4.1.1.3.2

Multiply $x$ by $1$ .

$x = - \frac{3}{16} \cdot - 1400$

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Simplify $- \frac{3}{16} \cdot - 1400$ .

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Step 6.4.2.1.1

Cancel the common factor of $8$ .

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Step 6.4.2.1.1.1

Move the leading negative in $- \frac{3}{16}$ into the numerator.

$x = \frac{- 3}{16} \cdot - 1400$

Step 6.4.2.1.1.2

Factor $8$ out of $16$ .

$x = \frac{- 3}{8 (2)} \cdot - 1400$

Step 6.4.2.1.1.3

Factor $8$ out of $- 1400$ .

$x = \frac{- 3}{8 \cdot 2} \cdot (8 \cdot - 175)$

Step 6.4.2.1.1.4

Cancel the common factor.

$x = \frac{- 3}{8 \cdot 2} \cdot (8 \cdot - 175)$

Step 6.4.2.1.1.5

Rewrite the expression.

$x = \frac{- 3}{2} \cdot - 175$

Step 6.4.2.1.2

Combine $\frac{- 3}{2}$ and $- 175$ .

$x = \frac{- 3 \cdot - 175}{2}$

Step 6.4.2.1.3

Multiply $- 3$ by $- 175$ .

$x = \frac{525}{2}$

Step 7

Find the values where the derivative is undefined.

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Step 7.1

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 8

Critical points to evaluate.

$x = \frac{525}{2}$

Step 9

Evaluate the second derivative at $x = \frac{525}{2}$ . If the second derivative is positive, then this is a local minimum. If it is negative, then this is a local maximum.

$- \frac{16}{3}$

Step 10

$x = \frac{525}{2}$ is a local maximum because the value of the second derivative is negative. This is referred to as the second derivative test.

$x = \frac{525}{2}$ is a local maximum

Step 11

Find the y-value when $x = \frac{525}{2}$ .

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Step 11.1

Replace the variable $x$ with $\frac{525}{2}$ in the expression.

$f (\frac{525}{2}) = 2 (700 - \frac{4 (\frac{525}{2})}{3}) (\frac{525}{2})$

Step 11.2

Simplify the result.

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Step 11.2.1

Simplify terms.

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Step 11.2.1.1

Cancel the common factor of $2$ .

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Step 11.2.1.1.1

Cancel the common factor.

$f (\frac{525}{2}) = 2 (700 - \frac{4 (\frac{525}{2})}{3}) (\frac{525}{2})$

Step 11.2.1.1.2

Rewrite the expression.

$f (\frac{525}{2}) = (700 - \frac{4 (\frac{525}{2})}{3}) \cdot 525$

Step 11.2.1.2

Combine $4$ and $\frac{525}{2}$ .

$f (\frac{525}{2}) = (700 - \frac{\frac{4 \cdot 525}{2}}{3}) \cdot 525$

Step 11.2.1.3

Multiply $4$ by $525$ .

$f (\frac{525}{2}) = (700 - \frac{\frac{2100}{2}}{3}) \cdot 525$

Step 11.2.2

Simplify each term.

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Step 11.2.2.1

Cancel the common factor of $2100$ and $2$ .

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Step 11.2.2.1.1

Factor $2$ out of $2100$ .

$f (\frac{525}{2}) = (700 - \frac{\frac{2 \cdot 1050}{2}}{3}) \cdot 525$

Step 11.2.2.1.2

Cancel the common factors.

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Step 11.2.2.1.2.1

Factor $2$ out of $2$ .

$f (\frac{525}{2}) = (700 - \frac{\frac{2 \cdot 1050}{2 (1)}}{3}) \cdot 525$

Step 11.2.2.1.2.2

Cancel the common factor.

$f (\frac{525}{2}) = (700 - \frac{\frac{2 \cdot 1050}{2 \cdot 1}}{3}) \cdot 525$

Step 11.2.2.1.2.3

Rewrite the expression.

$f (\frac{525}{2}) = (700 - \frac{\frac{1050}{1}}{3}) \cdot 525$

Step 11.2.2.1.2.4

Divide $1050$ by $1$ .

$f (\frac{525}{2}) = (700 - \frac{1050}{3}) \cdot 525$

Step 11.2.2.2

Cancel the common factor of $1050$ and $3$ .

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Step 11.2.2.2.1

Factor $3$ out of $1050$ .

$f (\frac{525}{2}) = (700 - \frac{3 \cdot 350}{3}) \cdot 525$

Step 11.2.2.2.2

Cancel the common factors.

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Step 11.2.2.2.2.1

Factor $3$ out of $3$ .

$f (\frac{525}{2}) = (700 - \frac{3 \cdot 350}{3 (1)}) \cdot 525$

Step 11.2.2.2.2.2

Cancel the common factor.

$f (\frac{525}{2}) = (700 - \frac{3 \cdot 350}{3 \cdot 1}) \cdot 525$

Step 11.2.2.2.2.3

Rewrite the expression.

$f (\frac{525}{2}) = (700 - \frac{350}{1}) \cdot 525$

Step 11.2.2.2.2.4

Divide $350$ by $1$ .

$f (\frac{525}{2}) = (700 - 1 \cdot 350) \cdot 525$

Step 11.2.2.3

Multiply $- 1$ by $350$ .

$f (\frac{525}{2}) = (700 - 350) \cdot 525$

Step 11.2.3

Simplify the expression.

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Step 11.2.3.1

Subtract $350$ from $700$ .

$f (\frac{525}{2}) = 350 \cdot 525$

Step 11.2.3.2

Multiply $350$ by $525$ .

$f (\frac{525}{2}) = 183750$

Step 11.2.4

The final answer is $183750$ .

$y = 183750$

Step 12

These are the local extrema for $f (x) = 2 (700 - \frac{4 x}{3}) x$ .

$(\frac{525}{2}, 183750)$ is a local maxima

Step 13