Find the critical numbers of the function

Question

f(x)=x^1/6−x^−5/6.

Answer 1

I will guess you meant:
f(x) = x^(1/6) - x^(-5/6)
then
f'(x) = (1/6)x^(-5/6 + (5/6)x^(-11/6)
= 0 for critical values

factor it ..
(1/6)x^(-11/6)[ x + 5] = 0
x = 0 or x = - 5

sub those values back into the original function if you need the y values