Let f(x)=x−x^1/4 on [0,1].

Question

Let f(x)=x−x^1/4 on [0,1].

f(0)=_________ f(1)=_________

Find c such that f'(c)=0 or, if Rolle's Theorem does not apply, enter DNE

c=_________ 

Answer 1

f(0) = 0

f(1) =0

f(x) is continuous on [0, 1] and differentiable on (0,1).

Thus, there exists a c in (0, 1) such that f'(c) = 0

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

1 = 1/4 x^-3/4

4 ⁼ x^-3/4

4^-4/3 = x

1/4^4/3 = x

1/2^8/3 = x

2^1/3/8 =x