What are the critical values, if any, of #f(x)= x^(3/4) - 2x^(1/4)#?

1 Answer
Feb 14, 2016

Answer:

By the definition I am accustomed to, they are #0# and #4/9#.

Explanation:

A critical value of #f# is a value in the domain of #f# at which #f'# does not exists or #f'(x)# is #0#.

The domain of the function #f(x)=x^(3/4)-2x^(1/4)# is #[0,oo)#.

The derivative is #f'(x) =3/4x^(-1/4)-1/2x^(-3/4) = (3x^(1/2)-2)/(4x^(3/2))#

#f'# fails to exist at #x=0# and #f'(x) = 0# at #x=4/9#.

Both #0# and #4/9# are in the domain, so both are critical values.