Find the critical numbers for f(x)=x^(4/5)(x-8)^2.?

Question

Do NOT round the answers.

Answer 1

f(x) = x^(4/5) (x-8)^2

f'(x) = (4/5) x^(-1/5) (x-8)^2 + x^(4/5) . 2 (x-8)
= x^(-1/5) . (x-8) . [(4/5) (x-8) + 2x]
This is undefined at x=0, zero at x=8 and when 4(x-8) + 10x = 0 => x = 16/7.

Corresponding values of f are 0, 0 and
(16/7)^(4/5) (16/7 - 8)^2
= [(2^4)^(4/5) / 7^(4/5)] . [40^2 / 7^2]
= [2^(16/5) . 40^2] / [7^(4/5) . 7^2]
= [1600 . 8 . 5th√2 . 5th√7] / [7^3]
= 12800 5th√14 / 343

where I am using 5th√ to represent the fifth root.