Question

49. Let \( f: R^{+} \rightarrow R \) be a differentiable function with \( f(1)=3 \) and satisfying : \( \int_{1}^{x y} f(t) d t=y \int_{1}^{x} f(t) d t+x \int_{1}^{y} f(t) d t \forall x, y \in R^{+} \), then \( f(e)= \) (a) 3 (b) 4 (c) 1 (d) None of these

Answer 1

\(\int^{xy}_1f(t)dt=y\int^x_1f(t)dt+x\int^4_1f(t)dt\)

differentiable both sides w.r.t x, we get

⇒ \(\frac{dy}{dx}=0\)

⇒ y = constant 

∴ f(e) = 3