Find the Maximum/Minimum Value y=-x^2+5x-10

$y = - x^{2} + 5 x - 10$

Step 1

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 2

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

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

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

Remove parentheses.

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

Simplify $- \frac{5}{2 (- 1)}$ .

Multiply $2$ by $- 1$ .

$x = - \frac{5}{- 2}$

Move the negative in front of the fraction.

$x = - - \frac{5}{2}$

Multiply $- - \frac{5}{2}$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $\frac{5}{2}$ by $1$ .

$x = \frac{5}{2}$

Step 3

Evaluate $f (\frac{5}{2})$ .

Replace the variable $x$ with $\frac{5}{2}$ in the expression.

$f (\frac{5}{2}) = - {(\frac{5}{2})}^{2} + 5 (\frac{5}{2}) - 10$

Simplify the result.

Simplify each term.

Apply the product rule to $\frac{5}{2}$ .

$f (\frac{5}{2}) = - \frac{5^{2}}{2^{2}} + 5 (\frac{5}{2}) - 10$

Raise $5$ to the power of $2$ .

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

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

$f (\frac{5}{2}) = - \frac{25}{4} + 5 (\frac{5}{2}) - 10$

Multiply $5 (\frac{5}{2})$ .

Combine $5$ and $\frac{5}{2}$ .

$f (\frac{5}{2}) = - \frac{25}{4} + \frac{5 \cdot 5}{2} - 10$

Multiply $5$ by $5$ .

$f (\frac{5}{2}) = - \frac{25}{4} + \frac{25}{2} - 10$

Find the common denominator.

Multiply $\frac{25}{2}$ by $\frac{2}{2}$ .

$f (\frac{5}{2}) = - \frac{25}{4} + \frac{25}{2} \cdot \frac{2}{2} - 10$

Multiply $\frac{25}{2}$ by $\frac{2}{2}$ .

$f (\frac{5}{2}) = - \frac{25}{4} + \frac{25 \cdot 2}{2 \cdot 2} - 10$

Write $- 10$ as a fraction with denominator $1$ .

$f (\frac{5}{2}) = - \frac{25}{4} + \frac{25 \cdot 2}{2 \cdot 2} + \frac{- 10}{1}$

Multiply $\frac{- 10}{1}$ by $\frac{4}{4}$ .

$f (\frac{5}{2}) = - \frac{25}{4} + \frac{25 \cdot 2}{2 \cdot 2} + \frac{- 10}{1} \cdot \frac{4}{4}$

Multiply $\frac{- 10}{1}$ by $\frac{4}{4}$ .

$f (\frac{5}{2}) = - \frac{25}{4} + \frac{25 \cdot 2}{2 \cdot 2} + \frac{- 10 \cdot 4}{4}$

Multiply $2$ by $2$ .

$f (\frac{5}{2}) = - \frac{25}{4} + \frac{25 \cdot 2}{4} + \frac{- 10 \cdot 4}{4}$

Combine the numerators over the common denominator.

$f (\frac{5}{2}) = \frac{- 25 + 25 \cdot 2 - 10 \cdot 4}{4}$

Simplify each term.

Multiply $25$ by $2$ .

$f (\frac{5}{2}) = \frac{- 25 + 50 - 10 \cdot 4}{4}$

Multiply $- 10$ by $4$ .

$f (\frac{5}{2}) = \frac{- 25 + 50 - 40}{4}$

Simplify the expression.

Add $- 25$ and $50$ .

$f (\frac{5}{2}) = \frac{25 - 40}{4}$

Subtract $40$ from $25$ .

$f (\frac{5}{2}) = \frac{- 15}{4}$

Move the negative in front of the fraction.

$f (\frac{5}{2}) = - \frac{15}{4}$

The final answer is $- \frac{15}{4}$ .

$- \frac{15}{4}$

Step 4

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

$(\frac{5}{2}, - \frac{15}{4})$

Step 5

Find the Maximum/Minimum Value y=-x^2+5x-10

Related Questions

Support LCMGCF.com by disabling your adblocker.