Use the First Derivative Test to find the location of all local extrema for the function given below. Enter an exact answer. If there is more than one local maximum or local minimum, write each value of $x$ separated by a comma. If a local maximum or local minimum does not occur on the function, enter $\varnothing$ in the appropriate box.
\[
f(x)=2 x \sqrt{5-5 x}
\]

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 \( \boxed{\frac{2}{3}} \)