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: Understanding the Concept
Let \(h(x)=f(x)-g(x)\). Then \(h''(x)=f''(x)-g''(x)=0\), so \(h'\) is constant and \(h\) is a linear function.

Step 2: Find the constants
\(f'(1)=4\) and \(2g'(1)=4\Rightarrow g'(1)=2\). So
\[ h'(1)=4-2=2 \]
Since \(h'\) is constant, \(h'(x)=2\) for all \(x\). So \(h(x)=2x+c\).

Step 3: Use the second condition
\(f(2)=9\) and \(3g(2)=9\Rightarrow g(2)=3\). So \(h(2)=9-3=6\).
\[ 4+c=6\Rightarrow c=2 \]

Step 4: Evaluate at 4
\[ h(4)=2(4)+2=10 \]
This is option (B).

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