Question

A real differentiable function \(f\) satisfies \(f(x)+f(y)+2xy=f(x+y)\). Given \(f''(0)=0\), then \[ \int_0^{\pi/2} f(\sin x)\,dx = \]

• A. \(0\)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{\pi}{2} \)
• D. \( \pi \)

Hint:

Recognize identity pattern: \( (x+y)^2 \).

Answer 1

The Correct Option is B

Concept:

Step 1: Put \(y=0\): \[ f(x)+f(0)=f(x) \Rightarrow f(0)=0 \]

Step 2: Assume: \[ f(x)=x^2 \] Check: \[ x^2 + y^2 + 2xy = (x+y)^2 \]

Step 3: \[ \int_0^{\pi/2} \sin^2 x\,dx = \frac{\pi}{4} \]