Find the Maximum/Minimum Value y=0.0013x^2-0.38x+31.67

$y = 0.0013 x^{2} - 0.38 x + 31.67$

Step 1

The minimum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is positive, the minimum value of the function is $f (- \frac{b}{2 a})$ .

$f_{min}$ $x = a x^{2} + b x + c$ 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 $b$ .

$x = - \frac{- 0.38}{2 (0.0013)}$

Remove parentheses.

$x = - \frac{- 0.38}{2 (0.0013)}$

Simplify $- \frac{- 0.38}{2 (0.0013)}$ .

Multiply $2$ by $0.0013$ .

$x = - \frac{- 0.38}{0.0026}$

Divide $- 0.38$ by $0.0026$ .

$x = - - 146.\overline{153846}$

Multiply $- 1$ by $- 146.\overline{153846}$ .

$x = 146.\overline{153846}$

Step 3

Evaluate $f (146.\overline{153846})$ .

Replace the variable $x$ with $146.\overline{153846}$ in the expression.

$f (146.\overline{153846}) = 0.0013 {(146.\overline{153846})}^{2} - 0.38 \cdot 146.\overline{153846} + 31.67$

Simplify the result.

Simplify each term.

Raise $146.\overline{153846}$ to the power of $2$ .

$f (146.\overline{153846}) = 0.0013 \cdot 21360.94674556 - 0.38 \cdot 146.\overline{153846} + 31.67$

Multiply $0.0013$ by $21360.94674556$ .

$f (146.\overline{153846}) = 27.\overline{769230} - 0.38 \cdot 146.\overline{153846} + 31.67$

Multiply $- 0.38$ by $146.\overline{153846}$ .

$f (146.\overline{153846}) = 27.\overline{769230} - 55.\overline{538461} + 31.67$

Simplify by adding and subtracting.

Subtract $55.\overline{538461}$ from $27.\overline{769230}$ .

$f (146.\overline{153846}) = - 27.76923076 + 31.67$

Add $- 27.76923076$ and $31.67$ .

$f (146.\overline{153846}) = 3.90076923$

The final answer is $3.90076923$ .

$3.90076923$

Step 4

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

$(146.\overline{153846}, 3.90076923)$

Step 5

Find the Maximum/Minimum Value y=0.0013x^2-0.38x+31.67

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