Verify that the Mean Value Theorem can be applied to the function f(x)=x^3/4 on the interval [0,16]. Then find the value of c in the interval that satisfies the conclusion of the Mean Value Theorem.

Answer 1

f(x) is a power function with an even denominator in the exponent, which means f is continuous on [0 , ∞) and differentiable on (0 , ∞) so the MVT applies on [0 , 16].

f(0) = 0 ; f(16) = ⁴√16³ = 8

Avg rate of change of f on [0 , 16] = (8 - 0) / (16 - 0) = 1/2

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

x^-1/4 = 2/3

x^1/4 = 3/2

x = 81/16 = c , the x-value with 0 < c < 16 such that f^'(c) = avg. rate of change of f on [0 , 16].