Question

The value of c∈(0,2) satisfying the mean value theorem for the function f(x)=x(x-1)² on [0,2] is:

• A. \(\frac{3}{4}\)
• B. \(\frac{4}{3}\)
• C. \(\frac{1}{3}\)
• D. (2)/(3)

Hint:

Ignore endpoint solutions while applying MVT.

Answer 1

The Correct Option is D

Step 1: Mean Value Theorem: f'(c)=(f(2)-f(0))/(2-0)
Step 2: f(2)=2(1)²=2, f(0)=0⟹ (2)/(2)=1.
Step 3: Differentiate: f'(x)=3x²-4x+1
Step 4: Set f'(c)=1: 3c²-4c=0 ⟹ c(3c-4)=0
Step 5: Valid solution in (0,2) is c=(2)/(3).