Find the Maximum/Minimum Value f(x)=-16/1225x^2+7/5x+1.5

$f (x) = - \frac{16}{1225} x^{2} + \frac{7}{5} x + 1.5$

Step 1

Simplify each term.

Combine $x^{2}$ and $\frac{16}{1225}$ .

$f (x) = - \frac{x^{2} \cdot 16}{1225} + \frac{7}{5} x + 1.5$

Move $16$ to the left of $x^{2}$ .

$f (x) = - \frac{16 \cdot x^{2}}{1225} + \frac{7}{5} x + 1.5$

Combine $\frac{7}{5}$ and $x$ .

$f (x) = - \frac{16 x^{2}}{1225} + \frac{7 x}{5} + 1.5$

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

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

Step 3

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

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

$x = - \frac{1.4}{2 (- 0.01306122)}$

Remove parentheses.

$x = - \frac{1.4}{2 (- 0.01306122)}$

Simplify $- \frac{1.4}{2 (- 0.01306122)}$ .

Multiply $2$ by $- 0.01306122$ .

$x = - \frac{1.4}{- 0.02612244}$

Divide $1.4$ by $- 0.02612244$ .

$x = - - 53.59375$

Multiply $- 1$ by $- 53.59375$ .

$x = 53.59375$

Step 4

Evaluate $f (53.59375)$ .

Replace the variable $x$ with $53.59375$ in the expression.

$f (53.59375) = - \frac{16 {(53.59375)}^{2}}{1225} + \frac{7 (53.59375)}{5} + 1.5$

Simplify the result.

Multiply $7$ by $53.59375$ .

$f (53.59375) = - \frac{16 {(53.59375)}^{2}}{1225} + \frac{375.15625}{5} + 1.5$

Simplify each term.

Raise $53.59375$ to the power of $2$ .

$f (53.59375) = - \frac{16 \cdot 2872.29003906}{1225} + \frac{375.15625}{5} + 1.5$

Multiply $16$ by $2872.29003906$ .

$f (53.59375) = - \frac{45956.640625}{1225} + \frac{375.15625}{5} + 1.5$

Divide $45956.640625$ by $1225$ .

$f (53.59375) = - 1 \cdot 37.515625 + \frac{375.15625}{5} + 1.5$

Multiply $- 1$ by $37.515625$ .

$f (53.59375) = - 37.515625 + \frac{375.15625}{5} + 1.5$

Divide $375.15625$ by $5$ .

$f (53.59375) = - 37.515625 + 75.03125 + 1.5$

Simplify by adding numbers.

Add $- 37.515625$ and $75.03125$ .

$f (53.59375) = 37.515625 + 1.5$

Add $37.515625$ and $1.5$ .

$f (53.59375) = 39.015625$

The final answer is $39.015625$ .

$39.015625$

Step 5

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

$(53.59375, 39.015625)$

Step 6

Find the Maximum/Minimum Value f(x)=-16/1225x^2+7/5x+1.5

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