Question

Let $f(x)$ and $g(x)$ be twice differentiable functions satisfying $f''(x) = g''(x)$ for all $x \in \mathbb{R}$, $f'(1) = 2g'(1) = 4$ and $g(2) = 3f(2) = 9$. Then $f(25) - g(25)$ is equal to :

• A. 20
• B. 40
• C. $-20$
• D. $-40$

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Integrate the double derivative equality twice to find the relationship between $f(x)$ and $g(x)$, then use given conditions to find constants.
Step 2: Detailed Explanation:
Given $f''(x) = g''(x)$. Integrating once:
$f'(x) = g'(x) + c_1$.
At $x=1, f'(1) = 4$ and $g'(1) = 2$.
$4 = 2 + c_1 \implies c_1 = 2$.
So $f'(x) = g'(x) + 2$. Integrating again:
$f(x) = g(x) + 2x + c_2 \implies f(x) - g(x) = 2x + c_2$.
At $x=2, g(2) = 9$ and $3f(2) = 9 \implies f(2) = 3$.
$3 - 9 = 2(2) + c_2 \implies -6 = 4 + c_2 \implies c_2 = -10$.
The general relation is $f(x) - g(x) = 2x - 10$.
At $x=25$, $f(25) - g(25) = 2(25) - 10 = 50 - 10 = 40$.
Step 3: Final Answer:
The value is 40.