Question

If \(f(x),g(x)\) be twice differentiable functions, satisfying \(f^{''}(x) = g^{''}(x),f^'(1) = 2g^'(1) = 4\) and \(f(2) = 3g(2) = 9\) then \(f(x)-g(x)\) at \(x = 4\) is equal to

• A. \(0\)
• B. \(10\)
• C. \(8\)
• D. \(2\)

Hint:

Equal second derivatives mean the difference of the two functions is a linear function.

Answer 1

The Correct Option is B

Step 1: Approach
Integrate the condition on second derivatives twice.

Step 2: First integration
$f''-g''=0$ gives $f'-g'=k$. At $x=1$: $k=4-2=2$.

Step 3: Second integration
$f-g=2x+c$. At $x=2$: $f(2)-g(2)=9-3=6$, so $4+c=6$ and $c=2$.

Step 4: Value
$f(4)-g(4)=8+2=10$. Option (B).

Final Answer:
The difference f - g equals 2x + 2, so at x = 4 it is 10, option (B). \[ \boxed{10} \]