Find the Maximum/Minimum Value -16t^2+62t+2

$- 16 t^{2} + 62 t + 2$

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 $t = - \frac{b}{2 a}$

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

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

$t = - \frac{62}{2 (- 16)}$

Remove parentheses.

$t = - \frac{62}{2 (- 16)}$

Simplify $- \frac{62}{2 (- 16)}$ .

Cancel the common factor of $62$ and $2$ .

Factor $2$ out of $62$ .

$t = - \frac{2 \cdot 31}{2 \cdot - 16}$

Cancel the common factors.

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

$t = - \frac{2 \cdot 31}{2 (- 16)}$

Cancel the common factor.

$t = - \frac{2 \cdot 31}{2 \cdot - 16}$

Rewrite the expression.

$t = - \frac{31}{- 16}$

Move the negative in front of the fraction.

$t = - - \frac{31}{16}$

Multiply $- - \frac{31}{16}$ .

Multiply $- 1$ by $- 1$ .

$t = 1 (\frac{31}{16})$

Multiply $\frac{31}{16}$ by $1$ .

$t = \frac{31}{16}$

Evaluate $f (\frac{31}{16})$ .

Replace the variable $t$ with $\frac{31}{16}$ in the expression.

$f (\frac{31}{16}) = - 16 {(\frac{31}{16})}^{2} + 62 (\frac{31}{16}) + 2$

Simplify the result.

Simplify each term.

Apply the product rule to $\frac{31}{16}$ .

$f (\frac{31}{16}) = - 16 \frac{31^{2}}{16^{2}} + 62 (\frac{31}{16}) + 2$

Raise $31$ to the power of $2$ .

$f (\frac{31}{16}) = - 16 \frac{961}{16^{2}} + 62 (\frac{31}{16}) + 2$

Raise $16$ to the power of $2$ .

$f (\frac{31}{16}) = - 16 (\frac{961}{256}) + 62 (\frac{31}{16}) + 2$

Cancel the common factor of $16$ .

Factor $16$ out of $- 16$ .

$f (\frac{31}{16}) = 16 (- 1) (\frac{961}{256}) + 62 (\frac{31}{16}) + 2$

Factor $16$ out of $256$ .

$f (\frac{31}{16}) = 16 \cdot (- 1 \frac{961}{16 \cdot 16}) + 62 (\frac{31}{16}) + 2$

Cancel the common factor.

$f (\frac{31}{16}) = 16 \cdot (- 1 \frac{961}{16 \cdot 16}) + 62 (\frac{31}{16}) + 2$

Rewrite the expression.

$f (\frac{31}{16}) = - 1 (\frac{961}{16}) + 62 (\frac{31}{16}) + 2$

Rewrite $- 1 (\frac{961}{16})$ as $- (\frac{961}{16})$ .

$f (\frac{31}{16}) = - (\frac{961}{16}) + 62 (\frac{31}{16}) + 2$

Cancel the common factor of $2$ .

Factor $2$ out of $62$ .

$f (\frac{31}{16}) = - \frac{961}{16} + 2 (31) (\frac{31}{16}) + 2$

Factor $2$ out of $16$ .

$f (\frac{31}{16}) = - \frac{961}{16} + 2 \cdot (31 (\frac{31}{2 \cdot 8})) + 2$

Cancel the common factor.

$f (\frac{31}{16}) = - \frac{961}{16} + 2 \cdot (31 (\frac{31}{2 \cdot 8})) + 2$

Rewrite the expression.

$f (\frac{31}{16}) = - \frac{961}{16} + 31 (\frac{31}{8}) + 2$

Combine $31$ and $\frac{31}{8}$ .

$f (\frac{31}{16}) = - \frac{961}{16} + \frac{31 \cdot 31}{8} + 2$

Multiply $31$ by $31$ .

$f (\frac{31}{16}) = - \frac{961}{16} + \frac{961}{8} + 2$

Find the common denominator.

Multiply $\frac{961}{8}$ by $\frac{2}{2}$ .

$f (\frac{31}{16}) = - \frac{961}{16} + \frac{961}{8} \cdot \frac{2}{2} + 2$

Multiply $\frac{961}{8}$ by $\frac{2}{2}$ .

$f (\frac{31}{16}) = - \frac{961}{16} + \frac{961 \cdot 2}{8 \cdot 2} + 2$

Write $2$ as a fraction with denominator $1$ .

$f (\frac{31}{16}) = - \frac{961}{16} + \frac{961 \cdot 2}{8 \cdot 2} + \frac{2}{1}$

Multiply $\frac{2}{1}$ by $\frac{16}{16}$ .

$f (\frac{31}{16}) = - \frac{961}{16} + \frac{961 \cdot 2}{8 \cdot 2} + \frac{2}{1} \cdot \frac{16}{16}$

Multiply $\frac{2}{1}$ by $\frac{16}{16}$ .

$f (\frac{31}{16}) = - \frac{961}{16} + \frac{961 \cdot 2}{8 \cdot 2} + \frac{2 \cdot 16}{16}$

Reorder the factors of $8 \cdot 2$ .

$f (\frac{31}{16}) = - \frac{961}{16} + \frac{961 \cdot 2}{2 \cdot 8} + \frac{2 \cdot 16}{16}$

Multiply $2$ by $8$ .

$f (\frac{31}{16}) = - \frac{961}{16} + \frac{961 \cdot 2}{16} + \frac{2 \cdot 16}{16}$

Combine the numerators over the common denominator.

$f (\frac{31}{16}) = \frac{- 961 + 961 \cdot 2 + 2 \cdot 16}{16}$

Simplify each term.

Multiply $961$ by $2$ .

$f (\frac{31}{16}) = \frac{- 961 + 1922 + 2 \cdot 16}{16}$

Multiply $2$ by $16$ .

$f (\frac{31}{16}) = \frac{- 961 + 1922 + 32}{16}$

Simplify by adding numbers.

Add $- 961$ and $1922$ .

$f (\frac{31}{16}) = \frac{961 + 32}{16}$

Add $961$ and $32$ .

$f (\frac{31}{16}) = \frac{993}{16}$

The final answer is $\frac{993}{16}$ .

$\frac{993}{16}$

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

$(\frac{31}{16}, \frac{993}{16})$

Find the Maximum/Minimum Value -16t^2+62t+2

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