Question

Let \(f(x)\) and \(g(x)\) be twice differentiable functions defined on \([0,2]\) such that \(f''(x) - g''(x) = 0\), \(f'(1)=4,\ g'(1)=2,\ f(2)=9,\ g(2)=3\). At \(x=\frac{3}{2}\), \(f(x)-g(x)\) is

• A. $2$
• B. $3$
• C. $5$
• D. $8$
• E. $10$

Hint:

If second derivatives are equal, their difference is always a linear function.

Answer 1

The Correct Option is C

Concept: If second derivatives are equal: \[ f''(x) = g''(x) \Rightarrow f'(x) - g'(x) = \text{constant} \Rightarrow f(x) - g(x) = \text{linear function} \]

Step 1: Integrate given condition
\[ f''(x) - g''(x) = 0 \Rightarrow f'(x) - g'(x) = C_1 \]

Step 2: Use given values
\[ f'(1) - g'(1) = 4 - 2 = 2 \Rightarrow C_1 = 2 \] \[ f'(x) - g'(x) = 2 \]

Step 3: Integrate again
\[ f(x) - g(x) = 2x + C_2 \]

Step 4: Find constant using $x=2$
\[ f(2) - g(2) = 9 - 3 = 6 \] \[ 6 = 2(2) + C_2 \Rightarrow C_2 = 2 \]

Step 5: Evaluate at $x=\frac{3}{2}$
\[ f(x) - g(x) = 2x + 2 \] \[ = 2\left(\frac{3}{2}\right) + 2 = 3 + 2 = 5 \] Final Conclusion:
Option (C)