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

1 Answer
Dec 1, 2015

Critical value is x=1/2

Explanation:

f(x)=x^(2/3)+x^(-1/3)

f'(x)=(2/3) x^(-1/3)-(1/3)x^(-4/3)

To find the critical points set f'(x)=0, and solve for x

( 2x^(-1/3)-x^(-4/3))/3=0

2x^(-1/3)-x^(-4/3) = 0

2x^(-1/3)=x^(-4/3) -> use law of exponential

2/(x^(1/3))=1/(x^(4/3)) -> cross multiplication

x^(4/3)/x^(1/3)=1/2

x=1/2