The derivative of f(x), differentiating by parts, is f'(x)=3(8-2x)/(2sqrt(x(8-x)))=3(4-x)/sqrt(x(8-x)).
When f'(x)=0 we have maximum or minimum values. When x=4, which is between the limits [0,8], f'(4 )=0 and f(4)=3sqrt(16)=12.
Putting in x=3 and 5 we can see that f(x)<12, so we have a maximum.
When x=0 and 8 f(x)=0, so the minimum value of f(x) is zero.