Question

Let $f(x)$ satisfy the functional equation $2f(x)+f(1-x)=x^2 \forall x \in R$, then $f'(5)$ is:

• A. 4
• B. 10
• C. 12
• D. 0

Answer 1

Answer: (A)

4

Answer 2

To find $f'(5)$, we first need to find the expression for $f(x)$.

1. Set up a system of equations:

   We are given: $2f(x) + f(1-x) = x^2$ ...(1)

   Replacing $x$ with $(1-x)$ in equation (1), we get: $2f(1-x) + f(1-(1-x)) = (1-x)^2$ $2f(1-x) + f(x) = (1-x)^2$ ...(2)

2. Solve for $f(x)$:

   From equation (1), we can express $f(1-x)$ as: $f(1-x) = x^2 - 2f(x)$

   Substitute this into equation (2): $f(x) + 2(x^2 - 2f(x)) = (1-x)^2$ $f(x) + 2x^2 - 4f(x) = 1 - 2x + x^2$ $-3f(x) = -x^2 - 2x + 1$ $f(x) = \frac{1}{3}(x^2 + 2x - 1)$

3. Differentiate $f(x)$:

   $f'(x) = \frac{d}{dx} \left( \frac{1}{3}(x^2 + 2x - 1) \right)$ $f'(x) = \frac{1}{3}(2x + 2)$

4. Evaluate $f'(5)$:

   $f'(5) = \frac{1}{3}(2(5) + 2)$ $f'(5) = \frac{1}{3}(12)$ $f'(5) = 4$