Let f:(0,∞)→R and F(x)=∫0x f(x)dt. If Fx2=x2(1+x) then f4 is equal to

A
5/4
B
7
C
4
D
2

Solution:

Given, $F (x^{2}) = x^{2} (1 + x) \Rightarrow \int_{0}^{x^{2}} f (t) dt = x^{2} (1 + x)$

On differentiating both sides w.r.t. r, we get

$\begin{array}{l} 2 xf (x^{2}) = 2 x + 3 x^{2} \\ \Rightarrow f (x^{2}) = 1 + \frac{3 x}{2} \\ \Rightarrow f (2^{2}) = 1 + \frac{3}{2} (2) = 4 \Rightarrow f (4) = 4 \end{array}$

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