Calculus Examples

$f (x) = x {(x - 10)}^{2}$ ; $[0, 10]$

Step 1

Find the critical points.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Find the first derivative.

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Step 1.1.1.1

Rewrite ${(x - 10)}^{2}$ as $(x - 10) (x - 10)$ .

$\frac{d}{d x} [x ((x - 10) (x - 10))]$

Step 1.1.1.2

Expand $(x - 10) (x - 10)$ using the FOIL Method.

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Step 1.1.1.2.1

Apply the distributive property.

$\frac{d}{d x} [x (x (x - 10) - 10 (x - 10))]$

Step 1.1.1.2.2

Apply the distributive property.

$\frac{d}{d x} [x (x \cdot x + x \cdot - 10 - 10 (x - 10))]$

Step 1.1.1.2.3

Apply the distributive property.

$\frac{d}{d x} [x (x \cdot x + x \cdot - 10 - 10 x - 10 \cdot - 10)]$

Step 1.1.1.3

Simplify and combine like terms.

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Step 1.1.1.3.1

Simplify each term.

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Step 1.1.1.3.1.1

Multiply $x$ by $x$ .

$\frac{d}{d x} [x (x^{2} + x \cdot - 10 - 10 x - 10 \cdot - 10)]$

Step 1.1.1.3.1.2

Move $- 10$ to the left of $x$ .

$\frac{d}{d x} [x (x^{2} - 10 \cdot x - 10 x - 10 \cdot - 10)]$

Step 1.1.1.3.1.3

Multiply $- 10$ by $- 10$ .

$\frac{d}{d x} [x (x^{2} - 10 x - 10 x + 100)]$

Step 1.1.1.3.2

Subtract $10 x$ from $- 10 x$ .

$\frac{d}{d x} [x (x^{2} - 20 x + 100)]$

Step 1.1.1.4

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 $f (x) = x$ and $g (x) = x^{2} - 20 x + 100$ .

$x \frac{d}{d x} [x^{2} - 20 x + 100] + (x^{2} - 20 x + 100) \frac{d}{d x} [x]$

Step 1.1.1.5

Differentiate.

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Step 1.1.1.5.1

By the Sum Rule, the derivative of $x^{2} - 20 x + 100$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [100]$ .

$x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [100]) + (x^{2} - 20 x + 100) \frac{d}{d x} [x]$

Step 1.1.1.5.2

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

$x (2 x + \frac{d}{d x} [- 20 x] + \frac{d}{d x} [100]) + (x^{2} - 20 x + 100) \frac{d}{d x} [x]$

Step 1.1.1.5.3

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

$x (2 x - 20 \frac{d}{d x} [x] + \frac{d}{d x} [100]) + (x^{2} - 20 x + 100) \frac{d}{d x} [x]$

Step 1.1.1.5.4

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

$x (2 x - 20 \cdot 1 + \frac{d}{d x} [100]) + (x^{2} - 20 x + 100) \frac{d}{d x} [x]$

Step 1.1.1.5.5

Multiply $- 20$ by $1$ .

$x (2 x - 20 + \frac{d}{d x} [100]) + (x^{2} - 20 x + 100) \frac{d}{d x} [x]$

Step 1.1.1.5.6

Since $100$ is constant with respect to $x$ , the derivative of $100$ with respect to $x$ is $0$ .

$x (2 x - 20 + 0) + (x^{2} - 20 x + 100) \frac{d}{d x} [x]$

Step 1.1.1.5.7

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

$x (2 x - 20) + (x^{2} - 20 x + 100) \frac{d}{d x} [x]$

Step 1.1.1.5.8

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

$x (2 x - 20) + (x^{2} - 20 x + 100) \cdot 1$

Step 1.1.1.5.9

Multiply $x^{2} - 20 x + 100$ by $1$ .

$x (2 x - 20) + x^{2} - 20 x + 100$

Step 1.1.1.6

Simplify.

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Step 1.1.1.6.1

Apply the distributive property.

$x (2 x) + x \cdot - 20 + x^{2} - 20 x + 100$

Step 1.1.1.6.2

Combine terms.

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Step 1.1.1.6.2.1

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

$2 (x^{1} x) + x \cdot - 20 + x^{2} - 20 x + 100$

Step 1.1.1.6.2.2

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

$2 (x^{1} x^{1}) + x \cdot - 20 + x^{2} - 20 x + 100$

Step 1.1.1.6.2.3

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

$2 x^{1 + 1} + x \cdot - 20 + x^{2} - 20 x + 100$

Step 1.1.1.6.2.4

Add $1$ and $1$ .

$2 x^{2} + x \cdot - 20 + x^{2} - 20 x + 100$

Step 1.1.1.6.2.5

Move $- 20$ to the left of $x$ .

$2 x^{2} - 20 \cdot x + x^{2} - 20 x + 100$

Step 1.1.1.6.2.6

Add $2 x^{2}$ and $x^{2}$ .

$3 x^{2} - 20 x - 20 x + 100$

Step 1.1.1.6.2.7

Subtract $20 x$ from $- 20 x$ .

$f' (x) = 3 x^{2} - 40 x + 100$

Step 1.1.2

The first derivative of $f (x)$ with respect to $x$ is $3 x^{2} - 40 x + 100$ .

$3 x^{2} - 40 x + 100$

Step 1.2

Set the first derivative equal to $0$ then solve the equation $3 x^{2} - 40 x + 100 = 0$ .

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Step 1.2.1

Set the first derivative equal to $0$ .

$3 x^{2} - 40 x + 100 = 0$

Step 1.2.2

Factor by grouping.

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Step 1.2.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot 100 = 300$ and whose sum is $b = - 40$ .

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Step 1.2.2.1.1

Factor $- 40$ out of $- 40 x$ .

$3 x^{2} - 40 x + 100 = 0$

Step 1.2.2.1.2

Rewrite $- 40$ as $- 10$ plus $- 30$

$3 x^{2} + (- 10 - 30) x + 100 = 0$

Step 1.2.2.1.3

Apply the distributive property.

$3 x^{2} - 10 x - 30 x + 100 = 0$

Step 1.2.2.2

Factor out the greatest common factor from each group.

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Step 1.2.2.2.1

Group the first two terms and the last two terms.

$(3 x^{2} - 10 x) - 30 x + 100 = 0$

Step 1.2.2.2.2

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

$x (3 x - 10) - 10 (3 x - 10) = 0$

Step 1.2.2.3

Factor the polynomial by factoring out the greatest common factor, $3 x - 10$ .

$(3 x - 10) (x - 10) = 0$

Step 1.2.3

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

$3 x - 10 = 0$

$x - 10 = 0$

Step 1.2.4

Set $3 x - 10$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $3 x - 10$ equal to $0$ .

$3 x - 10 = 0$

Step 1.2.4.2

Solve $3 x - 10 = 0$ for $x$ .

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Step 1.2.4.2.1

Add $10$ to both sides of the equation.

$3 x = 10$

Step 1.2.4.2.2

Divide each term in $3 x = 10$ by $3$ and simplify.

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Step 1.2.4.2.2.1

Divide each term in $3 x = 10$ by $3$ .

$\frac{3 x}{3} = \frac{10}{3}$

Step 1.2.4.2.2.2

Simplify the left side.

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Step 1.2.4.2.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.4.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{10}{3}$

Step 1.2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{3}$

Step 1.2.5

Set $x - 10$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Step 1.2.5.2

Add $10$ to both sides of the equation.

$x = 10$

Step 1.2.6

The final solution is all the values that make $(3 x - 10) (x - 10) = 0$ true.

$x = \frac{10}{3}, 10$

Step 1.3

Find the values where the derivative is undefined.

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Step 1.3.1

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 1.4

Evaluate $x {(x - 10)}^{2}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 1.4.1

Evaluate at $x = \frac{10}{3}$ .

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Step 1.4.1.1

Substitute $\frac{10}{3}$ for $x$ .

$(\frac{10}{3}) {((\frac{10}{3}) - 10)}^{2}$

Step 1.4.1.2

Simplify.

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Step 1.4.1.2.1

To write $- 10$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{10}{3} {(\frac{10}{3} - 10 \cdot \frac{3}{3})}^{2}$

Step 1.4.1.2.2

Combine $- 10$ and $\frac{3}{3}$ .

$\frac{10}{3} {(\frac{10}{3} + \frac{- 10 \cdot 3}{3})}^{2}$

Step 1.4.1.2.3

Combine the numerators over the common denominator.

$\frac{10}{3} {(\frac{10 - 10 \cdot 3}{3})}^{2}$

Step 1.4.1.2.4

Simplify the numerator.

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Step 1.4.1.2.4.1

Multiply $- 10$ by $3$ .

$\frac{10}{3} {(\frac{10 - 30}{3})}^{2}$

Step 1.4.1.2.4.2

Subtract $30$ from $10$ .

$\frac{10}{3} {(\frac{- 20}{3})}^{2}$

Step 1.4.1.2.5

Move the negative in front of the fraction.

$\frac{10}{3} {(- \frac{20}{3})}^{2}$

Step 1.4.1.2.6

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.4.1.2.6.1

Apply the product rule to $- \frac{20}{3}$ .

$\frac{10}{3} ({(- 1)}^{2} {(\frac{20}{3})}^{2})$

Step 1.4.1.2.6.2

Apply the product rule to $\frac{20}{3}$ .

$\frac{10}{3} ({(- 1)}^{2} \frac{20^{2}}{3^{2}})$

Step 1.4.1.2.7

Simplify the expression.

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Step 1.4.1.2.7.1

Raise $- 1$ to the power of $2$ .

$\frac{10}{3} (1 \frac{20^{2}}{3^{2}})$

Step 1.4.1.2.7.2

Multiply $\frac{20^{2}}{3^{2}}$ by $1$ .

$\frac{10}{3} \cdot \frac{20^{2}}{3^{2}}$

Step 1.4.1.2.8

Combine.

$\frac{10 \cdot 20^{2}}{3 \cdot 3^{2}}$

Step 1.4.1.2.9

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 1.4.1.2.9.1

Multiply $3$ by $3^{2}$ .

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Step 1.4.1.2.9.1.1

Raise $3$ to the power of $1$ .

$\frac{10 \cdot 20^{2}}{3^{1} \cdot 3^{2}}$

Step 1.4.1.2.9.1.2

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

$\frac{10 \cdot 20^{2}}{3^{1 + 2}}$

Step 1.4.1.2.9.2

Add $1$ and $2$ .

$\frac{10 \cdot 20^{2}}{3^{3}}$

Step 1.4.1.2.10

Raise $20$ to the power of $2$ .

$\frac{10 \cdot 400}{3^{3}}$

Step 1.4.1.2.11

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

$\frac{10 \cdot 400}{27}$

Step 1.4.1.2.12

Multiply $10$ by $400$ .

$\frac{4000}{27}$

Step 1.4.2

Evaluate at $x = 10$ .

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Step 1.4.2.1

Substitute $10$ for $x$ .

$(10) {((10) - 10)}^{2}$

Step 1.4.2.2

Simplify.

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Step 1.4.2.2.1

Subtract $10$ from $10$ .

$10 \cdot 0^{2}$

Step 1.4.2.2.2

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

$10 \cdot 0$

Step 1.4.2.2.3

Multiply $10$ by $0$ .

$0$

Step 1.4.3

List all of the points.

$(\frac{10}{3}, \frac{4000}{27}), (10, 0)$

Step 2

Evaluate at the included endpoints.

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Step 2.1

Evaluate at $x = 0$ .

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Step 2.1.1

Substitute $0$ for $x$ .

$(0) {((0) - 10)}^{2}$

Step 2.1.2

Simplify.

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Step 2.1.2.1

Subtract $10$ from $0$ .

$0 {(- 10)}^{2}$

Step 2.1.2.2

Raise $- 10$ to the power of $2$ .

$0 \cdot 100$

Step 2.1.2.3

Multiply $0$ by $100$ .

$0$

Step 2.2

Evaluate at $x = 10$ .

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Step 2.2.1

Substitute $10$ for $x$ .

$(10) {((10) - 10)}^{2}$

Step 2.2.2

Simplify.

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Step 2.2.2.1

Subtract $10$ from $10$ .

$10 \cdot 0^{2}$

Step 2.2.2.2

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

$10 \cdot 0$

Step 2.2.2.3

Multiply $10$ by $0$ .

$0$

Step 2.3

List all of the points.

$(0, 0), (10, 0)$

Step 3

Compare the $f (x)$ values found for each value of $x$ in order to determine the absolute maximum and minimum over the given interval. The maximum will occur at the highest $f (x)$ value and the minimum will occur at the lowest $f (x)$ value.

Absolute Maximum: $(\frac{10}{3}, \frac{4000}{27})$

Absolute Minimum: $(0, 0), (10, 0)$

Step 4