Calculus Examples

$y = - \frac{1}{5} {(x - 5)}^{2} - 2$

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$f (x) = - \frac{1}{5} \cdot ((x - 5) (x - 5)) - 2$

Step 1.2

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$f (x) = - \frac{1}{5} \cdot (x (x - 5) - 5 (x - 5)) - 2$

Step 1.2.2

Apply the distributive property.

$f (x) = - \frac{1}{5} \cdot (x \cdot x + x \cdot - 5 - 5 (x - 5)) - 2$

Step 1.2.3

Apply the distributive property.

$f (x) = - \frac{1}{5} \cdot (x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5) - 2$

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 $x$ by $x$ .

$f (x) = - \frac{1}{5} \cdot (x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5) - 2$

Step 1.3.1.2

Move $- 5$ to the left of $x$ .

$f (x) = - \frac{1}{5} \cdot (x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5) - 2$

Step 1.3.1.3

Multiply $- 5$ by $- 5$ .

$f (x) = - \frac{1}{5} \cdot (x^{2} - 5 x - 5 x + 25) - 2$

Step 1.3.2

Subtract $5 x$ from $- 5 x$ .

$f (x) = - \frac{1}{5} \cdot (x^{2} - 10 x + 25) - 2$

Step 1.4

Apply the distributive property.

$f (x) = - \frac{1}{5} x^{2} - \frac{1}{5} \cdot (- 10 x) - \frac{1}{5} \cdot 25 - 2$

Step 1.5

Simplify.

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Step 1.5.1

Combine $x^{2}$ and $\frac{1}{5}$ .

$f (x) = - \frac{x^{2}}{5} - \frac{1}{5} \cdot (- 10 x) - \frac{1}{5} \cdot 25 - 2$

Step 1.5.2

Cancel the common factor of $5$ .

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Step 1.5.2.1

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$f (x) = - \frac{x^{2}}{5} + \frac{- 1}{5} \cdot (- 10 x) - \frac{1}{5} \cdot 25 - 2$

Step 1.5.2.2

Factor $5$ out of $- 10 x$ .

$f (x) = - \frac{x^{2}}{5} + \frac{- 1}{5} \cdot (5 (- 2 x)) - \frac{1}{5} \cdot 25 - 2$

Step 1.5.2.3

Cancel the common factor.

$f (x) = - \frac{x^{2}}{5} + \frac{- 1}{5} \cdot (5 (- 2 x)) - \frac{1}{5} \cdot 25 - 2$

Step 1.5.2.4

Rewrite the expression.

$f (x) = - \frac{x^{2}}{5} - 1 (- 2 x) - \frac{1}{5} \cdot 25 - 2$

Step 1.5.3

Multiply $- 2$ by $- 1$ .

$f (x) = - \frac{x^{2}}{5} + 2 x - \frac{1}{5} \cdot 25 - 2$

Step 1.5.4

Cancel the common factor of $5$ .

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Step 1.5.4.1

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$f (x) = - \frac{x^{2}}{5} + 2 x + \frac{- 1}{5} \cdot 25 - 2$

Step 1.5.4.2

Factor $5$ out of $25$ .

$f (x) = - \frac{x^{2}}{5} + 2 x + \frac{- 1}{5} \cdot (5 (5)) - 2$

Step 1.5.4.3

Cancel the common factor.

$f (x) = - \frac{x^{2}}{5} + 2 x + \frac{- 1}{5} \cdot (5 \cdot 5) - 2$

Step 1.5.4.4

Rewrite the expression.

$f (x) = - \frac{x^{2}}{5} + 2 x - 1 \cdot 5 - 2$

Step 1.5.5

Multiply $- 1$ by $5$ .

$f (x) = - \frac{x^{2}}{5} + 2 x - 5 - 2$

Step 2

Subtract $2$ from $- 5$ .

$f (x) = - \frac{x^{2}}{5} + 2 x - 7$

Step 3

The maximum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is negative, the maximum value of the function is $f (- \frac{b}{2 a})$ .

$f_{max}$ $x = a x^{2} + b x + c$ occurs at $x = - \frac{b}{2 a}$

Step 4

Find the value of $x = - \frac{b}{2 a}$ .

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Step 4.1

Substitute in the values of $a$ and $b$ .

$x = - \frac{2}{2 (- 0.2)}$

Step 4.2

Remove parentheses.

$x = - \frac{2}{2 (- 0.2)}$

Step 4.3

Simplify $- \frac{2}{2 (- 0.2)}$ .

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Step 4.3.1

Cancel the common factor of $2$ .

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Step 4.3.1.1

Cancel the common factor.

$x = - \frac{2}{2 \cdot - 0.2}$

Step 4.3.1.2

Rewrite the expression.

$x = - \frac{1}{- 0.2}$

Step 4.3.2

Simplify the expression.

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Step 4.3.2.1

Divide $1$ by $- 0.2$ .

$x = - - 5$

Step 4.3.2.2

Multiply $- 1$ by $- 5$ .

$x = 5$

Step 5

Evaluate $f (5)$ .

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Step 5.1

Replace the variable $x$ with $5$ in the expression.

$f (5) = - \frac{{(5)}^{2}}{5} + 2 (5) - 7$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Cancel the common factor of ${(5)}^{2}$ and $5$ .

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Step 5.2.1.1.1

Factor $5$ out of ${(5)}^{2}$ .

$f (5) = - \frac{5 \cdot 5}{5} + 2 (5) - 7$

Step 5.2.1.1.2

Cancel the common factors.

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Step 5.2.1.1.2.1

Factor $5$ out of $5$ .

$f (5) = - \frac{5 \cdot 5}{5 (1)} + 2 (5) - 7$

Step 5.2.1.1.2.2

Cancel the common factor.

$f (5) = - \frac{5 \cdot 5}{5 \cdot 1} + 2 (5) - 7$

Step 5.2.1.1.2.3

Rewrite the expression.

$f (5) = - \frac{5}{1} + 2 (5) - 7$

Step 5.2.1.1.2.4

Divide $5$ by $1$ .

$f (5) = - 1 \cdot 5 + 2 (5) - 7$

Step 5.2.1.2

Multiply $- 1$ by $5$ .

$f (5) = - 5 + 2 (5) - 7$

Step 5.2.1.3

Multiply $2$ by $5$ .

$f (5) = - 5 + 10 - 7$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Add $- 5$ and $10$ .

$f (5) = 5 - 7$

Step 5.2.2.2

Subtract $7$ from $5$ .

$f (5) = - 2$

Step 5.2.3

The final answer is $- 2$ .

$- 2$

Step 6

Use the $x$ and $y$ values to find where the maximum occurs.

$(5, - 2)$

Step 7