verify Mean Value Theorem

Question

Verify that the Mean Value Theorem can be applied to the function f(x)=x^3/4 on the interval [0,1]. Then find the value of c in the interval that satisfies the conclusion of the Mean Value Theorem.

 

Enter the exact answer.

c=

 

c=

Answer 1

f(x)= x^3/4 = (x^3)/4

4y = x^3

on the interval [0,1]

y=0 for x=0

y = 1/4 for x=1

f(c) = c^3/4

f'(c) = 3c^2/4

f'(c) = (f(1)-f(0)]/(1-0) = 1/4= 3c^2/4

3c^2/4 = 1/4

3c^2 =1

c^2 = 1/3

c = sqr(1/3) = 1/sqr3

c= sqr3/3

= about 1.732.../3

= about .577350269189626

c= sqr3/3 = about .577