To find the critical values of the function [tex]\( f(x) = 5 \sqrt{x} - x^2 \)[/tex], we need to follow a multi-step procedure involving finding the derivative and solving for points where this derivative equals zero. Here is the detailed, step-by-step solution:
1. Define the function:
[tex]\( f(x) = 5 \sqrt{x} - x^2 \)[/tex]
2. Find the derivative of [tex]\( f(x) \)[/tex] with respect to [tex]\( x \)[/tex]:
To find the critical points, we first need to find the first derivative of the function:
[tex]\( f'(x) = \frac{d}{dx}(5 \sqrt{x} - x^2) \)[/tex]
For [tex]\( 5 \sqrt{x} \)[/tex], which is the same as [tex]\( 5 x^{1/2} \)[/tex], the derivative is given by using the power rule:
[tex]\[
\frac{d}{dx}(5 x^{1/2}) = 5 \cdot \frac{1}{2} x^{-1/2} = \frac{5}{2} x^{-1/2} = \frac{5}{2 \sqrt{x}}
\][/tex]
For [tex]\( -x^2 \)[/tex], the derivative is:
[tex]\[
\frac{d}{dx}(-x^2) = -2x
\][/tex]
Therefore, the derivative of the function [tex]\( f(x) \)[/tex] is:
[tex]\[
f'(x) = \frac{5}{2 \sqrt{x}} - 2x
\][/tex]
3. Set the derivative equal to zero to find critical points:
We find critical points by solving:
[tex]\[
\frac{5}{2 \sqrt{x}} - 2x = 0
\][/tex]
4. Isolate [tex]\( x \)[/tex]:
Multiply through by [tex]\( 2 \sqrt{x} \)[/tex] to eliminate the fraction:
[tex]\[
5 - 4x \sqrt{x} = 0
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
5 = 4x \sqrt{x}
\][/tex]
Dividing both sides by 4:
[tex]\[
\frac{5}{4} = x \sqrt{x}
\][/tex]
This can be rewritten as:
[tex]\[
\frac{5}{4} = x^{3/2}
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], we raise both sides of the equation to the power of [tex]\( \frac{2}{3} \)[/tex]:
[tex]\[
x = \left( \frac{5}{4} \right)^{2/3}
\][/tex]
Thus, the critical value of [tex]\( f(x) = 5 \sqrt{x} - x^2 \)[/tex] is given by:
[tex]\[
x = \left( \frac{5}{4} \right)^{2/3}
\][/tex]
However, to represent this in a simpler form, since the result given is [tex]\( 10^{2/3} / 4 \)[/tex], which simplifies to the same value:
[tex]\[
x = 10^{2/3} / 4
\][/tex]
Therefore, the critical value of [tex]\( f(x) \)[/tex] is:
[tex]\[
x = \frac{10^{2/3}}{4}
\][/tex]
Thus, the critical value(s) of [tex]\( f(x)=5 \sqrt{x}-x^2 \)[/tex] are at [tex]\( x = \boxed{\frac{10^{2/3}}{4}} \)[/tex].
