Step 1 :Find the first derivative of the function: \( f'(x) = 2 \sqrt{5-5x} - \frac{5x}{\sqrt{5-5x}} \)
Step 2 :Set the first derivative equal to zero and solve for \( x \): \( 2 \sqrt{5-5x} - \frac{5x}{\sqrt{5-5x}} = 0 \)
Step 3 :Simplify the equation: \( 2(5-5x) - 5x = 0 \)
Step 4 :Expand and rearrange the equation: \( 10 - 10x - 5x = 0 \)
Step 5 :Solve for \( x \): \( 15x = 10 \) \( x = \frac{2}{3} \)
Step 6 :Evaluate the sign of the first derivative on either side of the critical point:
Step 7 :For \( x_1 = 0 \), \( f'(x_1) = 2 \sqrt{5} > 0 \)
Step 8 :For \( x_2 = 1 \), \( f'(x_2) = 2 \sqrt{0} = 0 \)
Step 9 :Since the sign of the first derivative changes from positive to zero at \( x = \frac{2}{3} \), it is a local minimum.
Step 10 :The location of all local extrema is \( \boxed{\frac{2}{3}} \)