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Let $f : (0, \infty) \to R$ and $F (x) = \int_{0}^{x} f (t) d t .$ If $F (x^{2}) = x^{2} (1 + x)$ then $f (4)$ equals

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Correct option is C

Fx2=∫0x2f(t) dt , there fore, x2(1+x)=∫0x2f(t)dt. Differentiating both sides w.r.t. x using Property17, we have2x+3x2=fx2·2x⇒fx2=1+(3/2)xPutting x=2 we have f(4)=1+3=4.

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Let f:(0,∞)→R and F(x)=∫0xf(t)dt. If Fx2=x2(1+x) then f(4) equals

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Let $f : (0, \infty) \to R$ and $F (x) = \int_{0}^{x} f (t) d t .$ If $F (x^{2}) = x^{2} (1 + x)$ then $f (4)$ equals