Question

Determine if Rolle's Theorem can be applied to the function on the given interval; if so, find the values of c guaranteed by the theorem. f(x)= x^2/3 - 1 on [-8,8]

Answer 1

continuous and differential on [-8,8] and f(-8)=f(-8)
then take f'(x) = 0.

Answer 2

don't forget to do the first part. that is, don't forget to see if \(f\) is differentiable on \((-8,8)\) which means the derivative has to exist for all numbers in that interval
take the derivative, and check
hint: it isn't

Answer 3

I got 2/3x^1/3 = 3...?

Answer 4

exponent should be negative

Answer 5

what do you next to solve?

Answer 6

i don't think there is an answer cuz then 2/3x^1/3 =0 has no solution.

Answer 7

there is no answer because it doesn't satisfy the hypotheses
the derivative does not exist at \(x=0\)

Answer 8

the derivative is
\[\frac{2}{3\sqrt[3]{x}}\] and this is undefined at \(x=0\) so the function is not differentiable at zero

Answer 9

and therefore it violates the hypothesis of rolles theorem, which requires the function to be differentiable on the entire interval

Answer 10

thank you!(: