Find the Maximum/Minimum Value f(x)=-2(x+25)^2+2

$f (x) = - 2 {(x + 25)}^{2} + 2$

Step 1

Simplify each term.

Rewrite ${(x + 25)}^{2}$ as $(x + 25) (x + 25)$ .

$f (x) = - 2 ((x + 25) (x + 25)) + 2$

Expand $(x + 25) (x + 25)$ using the FOIL Method.

Apply the distributive property.

$f (x) = - 2 (x (x + 25) + 25 (x + 25)) + 2$

Apply the distributive property.

$f (x) = - 2 (x \cdot x + x \cdot 25 + 25 (x + 25)) + 2$

Apply the distributive property.

$f (x) = - 2 (x \cdot x + x \cdot 25 + 25 x + 25 \cdot 25) + 2$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x) = - 2 (x^{2} + x \cdot 25 + 25 x + 25 \cdot 25) + 2$

Move $25$ to the left of $x$ .

$f (x) = - 2 (x^{2} + 25 \cdot x + 25 x + 25 \cdot 25) + 2$

Multiply $25$ by $25$ .

$f (x) = - 2 (x^{2} + 25 x + 25 x + 625) + 2$

Add $25 x$ and $25 x$ .

$f (x) = - 2 (x^{2} + 50 x + 625) + 2$

Apply the distributive property.

$f (x) = - 2 x^{2} - 2 (50 x) - 2 \cdot 625 + 2$

Simplify.

Multiply $50$ by $- 2$ .

$f (x) = - 2 x^{2} - 100 x - 2 \cdot 625 + 2$

Multiply $- 2$ by $625$ .

$f (x) = - 2 x^{2} - 100 x - 1250 + 2$

Step 2

Add $- 1250$ and $2$ .

$f (x) = - 2 x^{2} - 100 x - 1248$

Step 3

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 4

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

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

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

Remove parentheses.

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

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

Cancel the common factor of $- 100$ and $2$ .

Factor $2$ out of $- 100$ .

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

Cancel the common factors.

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

$x = - \frac{2 \cdot - 50}{2 (- 2)}$

Cancel the common factor.

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

Rewrite the expression.

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

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

Factor $- 2$ out of $- 50$ .

$x = - \frac{- 2 (25)}{- 2}$

Cancel the common factors.

Factor $- 2$ out of $- 2$ .

$x = - \frac{- 2 \cdot 25}{- 2 \cdot 1}$

Cancel the common factor.

$x = - \frac{- 2 \cdot 25}{- 2 \cdot 1}$

Rewrite the expression.

$x = - \frac{25}{1}$

Divide $25$ by $1$ .

$x = - 1 \cdot 25$

Multiply $- 1$ by $25$ .

$x = - 25$

Step 5

Evaluate $f (- 25)$ .

Replace the variable $x$ with $- 25$ in the expression.

$f (- 25) = - 2 {(- 25)}^{2} - 100 \cdot - 25 - 1248$

Simplify the result.

Simplify each term.

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

$f (- 25) = - 2 \cdot 625 - 100 \cdot - 25 - 1248$

Multiply $- 2$ by $625$ .

$f (- 25) = - 1250 - 100 \cdot - 25 - 1248$

Multiply $- 100$ by $- 25$ .

$f (- 25) = - 1250 + 2500 - 1248$

Simplify by adding and subtracting.

Add $- 1250$ and $2500$ .

$f (- 25) = 1250 - 1248$

Subtract $1248$ from $1250$ .

$f (- 25) = 2$

The final answer is $2$ .

$2$

Step 6

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

$(- 25, 2)$

Step 7

Find the Maximum/Minimum Value f(x)=-2(x+25)^2+2

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