a.) Determine whether the Mean Value Theorem can be applied to 𝑓(𝑥) = 𝑥^2/3, [1,8]

Question

b.) If the MVT can be applied, find all values of 𝑐 in the open interval (a, b) such that 𝑓′(𝑐) = 𝑓(𝑏) ― 𝑓(𝑎)/𝑏 ― 𝑎 . (you can leave the answer as a fraction) If the MVT cannot be applied, enter NA.

Answer 1

f'(x) = 2/3x^-1/3. So, f(x) continuous and differentiable on inerval (1, 8); (f(8) - f(1))/(8 - 1) = (4 - 1)/(8 - 1) = 3/7;

f'(c) = 2/3c^-1/3 = 3/7; c = (3/7·3/2)^-3 = (14/9)³ = 2744/729