Find the Maximum/Minimum Value -16t^2+50t+5

$- 16 t^{2} + 50 t + 5$

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 $b$ .

$t = - \frac{50}{2 (- 16)}$

Remove parentheses.

$t = - \frac{50}{2 (- 16)}$

Simplify $- \frac{50}{2 (- 16)}$ .

Cancel the common factor of $50$ and $2$ .

Factor $2$ out of $50$ .

$t = - \frac{2 \cdot 25}{2 \cdot - 16}$

Cancel the common factors.

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

$t = - \frac{2 \cdot 25}{2 (- 16)}$

Cancel the common factor.

$t = - \frac{2 \cdot 25}{2 \cdot - 16}$

Rewrite the expression.

$t = - \frac{25}{- 16}$

Move the negative in front of the fraction.

$t = - - \frac{25}{16}$

Multiply $- - \frac{25}{16}$ .

Multiply $- 1$ by $- 1$ .

$t = 1 (\frac{25}{16})$

Multiply $\frac{25}{16}$ by $1$ .

$t = \frac{25}{16}$

Step 3

Evaluate $f (\frac{25}{16})$ .

Replace the variable $t$ with $\frac{25}{16}$ in the expression.

$f (\frac{25}{16}) = - 16 {(\frac{25}{16})}^{2} + 50 (\frac{25}{16}) + 5$

Simplify the result.

Simplify each term.

Apply the product rule to $\frac{25}{16}$ .

$f (\frac{25}{16}) = - 16 \frac{25^{2}}{16^{2}} + 50 (\frac{25}{16}) + 5$

Raise $25$ to the power of $2$ .

$f (\frac{25}{16}) = - 16 \frac{625}{16^{2}} + 50 (\frac{25}{16}) + 5$

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

$f (\frac{25}{16}) = - 16 (\frac{625}{256}) + 50 (\frac{25}{16}) + 5$

Cancel the common factor of $16$ .

Factor $16$ out of $- 16$ .

$f (\frac{25}{16}) = 16 (- 1) (\frac{625}{256}) + 50 (\frac{25}{16}) + 5$

Factor $16$ out of $256$ .

$f (\frac{25}{16}) = 16 \cdot (- 1 \frac{625}{16 \cdot 16}) + 50 (\frac{25}{16}) + 5$

Cancel the common factor.

$f (\frac{25}{16}) = 16 \cdot (- 1 \frac{625}{16 \cdot 16}) + 50 (\frac{25}{16}) + 5$

Rewrite the expression.

$f (\frac{25}{16}) = - 1 (\frac{625}{16}) + 50 (\frac{25}{16}) + 5$

Rewrite $- 1 (\frac{625}{16})$ as $- (\frac{625}{16})$ .

$f (\frac{25}{16}) = - (\frac{625}{16}) + 50 (\frac{25}{16}) + 5$

Cancel the common factor of $2$ .

Factor $2$ out of $50$ .

$f (\frac{25}{16}) = - \frac{625}{16} + 2 (25) (\frac{25}{16}) + 5$

Factor $2$ out of $16$ .

$f (\frac{25}{16}) = - \frac{625}{16} + 2 \cdot (25 (\frac{25}{2 \cdot 8})) + 5$

Cancel the common factor.

$f (\frac{25}{16}) = - \frac{625}{16} + 2 \cdot (25 (\frac{25}{2 \cdot 8})) + 5$

Rewrite the expression.

$f (\frac{25}{16}) = - \frac{625}{16} + 25 (\frac{25}{8}) + 5$

Combine $25$ and $\frac{25}{8}$ .

$f (\frac{25}{16}) = - \frac{625}{16} + \frac{25 \cdot 25}{8} + 5$

Multiply $25$ by $25$ .

$f (\frac{25}{16}) = - \frac{625}{16} + \frac{625}{8} + 5$

Find the common denominator.

Multiply $\frac{625}{8}$ by $\frac{2}{2}$ .

$f (\frac{25}{16}) = - \frac{625}{16} + \frac{625}{8} \cdot \frac{2}{2} + 5$

Multiply $\frac{625}{8}$ by $\frac{2}{2}$ .

$f (\frac{25}{16}) = - \frac{625}{16} + \frac{625 \cdot 2}{8 \cdot 2} + 5$

Write $5$ as a fraction with denominator $1$ .

$f (\frac{25}{16}) = - \frac{625}{16} + \frac{625 \cdot 2}{8 \cdot 2} + \frac{5}{1}$

Multiply $\frac{5}{1}$ by $\frac{16}{16}$ .

$f (\frac{25}{16}) = - \frac{625}{16} + \frac{625 \cdot 2}{8 \cdot 2} + \frac{5}{1} \cdot \frac{16}{16}$

Multiply $\frac{5}{1}$ by $\frac{16}{16}$ .

$f (\frac{25}{16}) = - \frac{625}{16} + \frac{625 \cdot 2}{8 \cdot 2} + \frac{5 \cdot 16}{16}$

Reorder the factors of $8 \cdot 2$ .

$f (\frac{25}{16}) = - \frac{625}{16} + \frac{625 \cdot 2}{2 \cdot 8} + \frac{5 \cdot 16}{16}$

Multiply $2$ by $8$ .

$f (\frac{25}{16}) = - \frac{625}{16} + \frac{625 \cdot 2}{16} + \frac{5 \cdot 16}{16}$

Combine the numerators over the common denominator.

$f (\frac{25}{16}) = \frac{- 625 + 625 \cdot 2 + 5 \cdot 16}{16}$

Simplify each term.

Multiply $625$ by $2$ .

$f (\frac{25}{16}) = \frac{- 625 + 1250 + 5 \cdot 16}{16}$

Multiply $5$ by $16$ .

$f (\frac{25}{16}) = \frac{- 625 + 1250 + 80}{16}$

Simplify by adding numbers.

Add $- 625$ and $1250$ .

$f (\frac{25}{16}) = \frac{625 + 80}{16}$

Add $625$ and $80$ .

$f (\frac{25}{16}) = \frac{705}{16}$

The final answer is $\frac{705}{16}$ .

$\frac{705}{16}$

Step 4

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

$(\frac{25}{16}, \frac{705}{16})$

Step 5

Find the Maximum/Minimum Value -16t^2+50t+5

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