Find the critical value(s) of [tex]f(x)=5 \sqrt{x}-x^2[/tex]. Separate multiple answers with a comma.

[tex]f(x)[/tex] has critical values at [tex]x= \square[/tex].



Answer :

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].