Find the Maximum/Minimum Value -16t^2+88t

$- 16 t^{2} + 88 t$

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 $f (- \frac{b}{2 a})$ .

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

Step 2

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

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

$t = - \frac{88}{2 (- 16)}$

Remove parentheses.

$t = - \frac{88}{2 (- 16)}$

Simplify $- \frac{88}{2 (- 16)}$ .

Cancel the common factor of $88$ and $2$ .

Factor $2$ out of $88$ .

$t = - \frac{2 \cdot 44}{2 \cdot - 16}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 16$ .

$t = - \frac{2 \cdot 44}{2 (- 16)}$

Cancel the common factor.

$t = - \frac{2 \cdot 44}{2 \cdot - 16}$

Rewrite the expression.

$t = - \frac{44}{- 16}$

Cancel the common factor of $44$ and $- 16$ .

Factor $4$ out of $44$ .

$t = - \frac{4 (11)}{- 16}$

Cancel the common factors.

Factor $4$ out of $- 16$ .

$t = - \frac{4 \cdot 11}{4 \cdot - 4}$

Cancel the common factor.

$t = - \frac{4 \cdot 11}{4 \cdot - 4}$

Rewrite the expression.

$t = - \frac{11}{- 4}$

Move the negative in front of the fraction.

$t = - - \frac{11}{4}$

Multiply $- - \frac{11}{4}$ .

Multiply $- 1$ by $- 1$ .

$t = 1 (\frac{11}{4})$

Multiply $\frac{11}{4}$ by $1$ .

$t = \frac{11}{4}$

Step 3

Evaluate $f (\frac{11}{4})$ .

Replace the variable $t$ with $\frac{11}{4}$ in the expression.

$f (\frac{11}{4}) = - 16 {(\frac{11}{4})}^{2} + 88 (\frac{11}{4})$

Simplify the result.

Simplify each term.

Apply the product rule to $\frac{11}{4}$ .

$f (\frac{11}{4}) = - 16 \frac{11^{2}}{4^{2}} + 88 (\frac{11}{4})$

Raise $11$ to the power of $2$ .

$f (\frac{11}{4}) = - 16 \frac{121}{4^{2}} + 88 (\frac{11}{4})$

Raise $4$ to the power of $2$ .

$f (\frac{11}{4}) = - 16 (\frac{121}{16}) + 88 (\frac{11}{4})$

Cancel the common factor of $16$ .

Factor $16$ out of $- 16$ .

$f (\frac{11}{4}) = 16 (- 1) (\frac{121}{16}) + 88 (\frac{11}{4})$

Cancel the common factor.

$f (\frac{11}{4}) = 16 \cdot (- 1 (\frac{121}{16})) + 88 (\frac{11}{4})$

Rewrite the expression.

$f (\frac{11}{4}) = - 1 \cdot 121 + 88 (\frac{11}{4})$

Multiply $- 1$ by $121$ .

$f (\frac{11}{4}) = - 121 + 88 (\frac{11}{4})$

Cancel the common factor of $4$ .

Factor $4$ out of $88$ .

$f (\frac{11}{4}) = - 121 + 4 (22) (\frac{11}{4})$

Cancel the common factor.

$f (\frac{11}{4}) = - 121 + 4 \cdot (22 (\frac{11}{4}))$

Rewrite the expression.

$f (\frac{11}{4}) = - 121 + 22 \cdot 11$

Multiply $22$ by $11$ .

$f (\frac{11}{4}) = - 121 + 242$

Add $- 121$ and $242$ .

$f (\frac{11}{4}) = 121$

The final answer is $121$ .

$121$

Step 4

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

$(\frac{11}{4}, 121)$

Step 5

Find the Maximum/Minimum Value -16t^2+88t

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