Find the Maximum/Minimum Value 100x^2-1500x

$100 x^{2} - 1500 x$

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 $b$ .

$x = - \frac{- 1500}{2 (100)}$

Remove parentheses.

$x = - \frac{- 1500}{2 (100)}$

Simplify $- \frac{- 1500}{2 (100)}$ .

Cancel the common factor of $- 1500$ and $2$ .

Factor $2$ out of $- 1500$ .

$x = - \frac{2 \cdot - 750}{2 \cdot 100}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 100$ .

$x = - \frac{2 \cdot - 750}{2 (100)}$

Cancel the common factor.

$x = - \frac{2 \cdot - 750}{2 \cdot 100}$

Rewrite the expression.

$x = - \frac{- 750}{100}$

Cancel the common factor of $- 750$ and $100$ .

Factor $50$ out of $- 750$ .

$x = - \frac{50 (- 15)}{100}$

Cancel the common factors.

Factor $50$ out of $100$ .

$x = - \frac{50 \cdot - 15}{50 \cdot 2}$

Cancel the common factor.

$x = - \frac{50 \cdot - 15}{50 \cdot 2}$

Rewrite the expression.

$x = - \frac{- 15}{2}$

Move the negative in front of the fraction.

$x = - - \frac{15}{2}$

Multiply $- - \frac{15}{2}$ .

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{15}{2})$

Multiply $\frac{15}{2}$ by $1$ .

$x = \frac{15}{2}$

Step 3

Evaluate $f (\frac{15}{2})$ .

Replace the variable $x$ with $\frac{15}{2}$ in the expression.

$f (\frac{15}{2}) = 100 {(\frac{15}{2})}^{2} - 1500 (\frac{15}{2})$

Simplify the result.

Simplify each term.

Apply the product rule to $\frac{15}{2}$ .

$f (\frac{15}{2}) = 100 (\frac{15^{2}}{2^{2}}) - 1500 (\frac{15}{2})$

Raise $15$ to the power of $2$ .

$f (\frac{15}{2}) = 100 (\frac{225}{2^{2}}) - 1500 (\frac{15}{2})$

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

$f (\frac{15}{2}) = 100 (\frac{225}{4}) - 1500 (\frac{15}{2})$

Cancel the common factor of $4$ .

Factor $4$ out of $100$ .

$f (\frac{15}{2}) = 4 (25) (\frac{225}{4}) - 1500 (\frac{15}{2})$

Cancel the common factor.

$f (\frac{15}{2}) = 4 \cdot (25 (\frac{225}{4})) - 1500 (\frac{15}{2})$

Rewrite the expression.

$f (\frac{15}{2}) = 25 \cdot 225 - 1500 (\frac{15}{2})$

Multiply $25$ by $225$ .

$f (\frac{15}{2}) = 5625 - 1500 (\frac{15}{2})$

Cancel the common factor of $2$ .

Factor $2$ out of $- 1500$ .

$f (\frac{15}{2}) = 5625 + 2 (- 750) (\frac{15}{2})$

Cancel the common factor.

$f (\frac{15}{2}) = 5625 + 2 \cdot (- 750 (\frac{15}{2}))$

Rewrite the expression.

$f (\frac{15}{2}) = 5625 - 750 \cdot 15$

Multiply $- 750$ by $15$ .

$f (\frac{15}{2}) = 5625 - 11250$

Subtract $11250$ from $5625$ .

$f (\frac{15}{2}) = - 5625$

The final answer is $- 5625$ .

$- 5625$

Step 4

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

$(\frac{15}{2}, - 5625)$

Step 5

Find the Maximum/Minimum Value 100x^2-1500x

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