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

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

Step 1

Simplify by multiplying through.

Apply the distributive property.

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

Reorder.

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

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

Rewrite using the commutative property of multiplication.

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

Step 2

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x$ .

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

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 $b$ .

$x = - \frac{25}{2 (- 1)}$

Remove parentheses.

$x = - \frac{25}{2 (- 1)}$

Simplify $- \frac{25}{2 (- 1)}$ .

Multiply $2$ by $- 1$ .

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

Move the negative in front of the fraction.

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

Multiply $- - \frac{25}{2}$ .

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{25}{2})$

Multiply $\frac{25}{2}$ by $1$ .

$x = \frac{25}{2}$

Step 5

Evaluate $f (\frac{25}{2})$ .

Replace the variable $x$ with $\frac{25}{2}$ in the expression.

$f (\frac{25}{2}) = 25 (\frac{25}{2}) - {(\frac{25}{2})}^{2}$

Simplify the result.

Simplify each term.

Multiply $25 (\frac{25}{2})$ .

Combine $25$ and $\frac{25}{2}$ .

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

Multiply $25$ by $25$ .

$f (\frac{25}{2}) = \frac{625}{2} - {(\frac{25}{2})}^{2}$

Apply the product rule to $\frac{25}{2}$ .

$f (\frac{25}{2}) = \frac{625}{2} - \frac{25^{2}}{2^{2}}$

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

$f (\frac{25}{2}) = \frac{625}{2} - \frac{625}{2^{2}}$

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

$f (\frac{25}{2}) = \frac{625}{2} - \frac{625}{4}$

To write $\frac{625}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (\frac{25}{2}) = \frac{625}{2} \cdot \frac{2}{2} - \frac{625}{4}$

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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

$f (\frac{25}{2}) = \frac{625 \cdot 2}{2 \cdot 2} - \frac{625}{4}$

Multiply $2$ by $2$ .

$f (\frac{25}{2}) = \frac{625 \cdot 2}{4} - \frac{625}{4}$

Combine the numerators over the common denominator.

$f (\frac{25}{2}) = \frac{625 \cdot 2 - 625}{4}$

Simplify the numerator.

Multiply $625$ by $2$ .

$f (\frac{25}{2}) = \frac{1250 - 625}{4}$

Subtract $625$ from $1250$ .

$f (\frac{25}{2}) = \frac{625}{4}$

The final answer is $\frac{625}{4}$ .

$\frac{625}{4}$

Step 6

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

$(\frac{25}{2}, \frac{625}{4})$

Step 7

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

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