Mean Value Theorem

Step 1: Apply Rolle’s Theorem.
Rolle's theorem can be applied when the function is continuous, differentiable, and the function values at the endpoints are equal. By solving for , we find .
Thus, the correct answer is (4).

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).