Let f(x) and g(x) are differentiable function of x such that x2(gf(x)).(f'g(x)).(g'(x))=(1−2x2)(g'f(x)).(f'(x))f(g(x))∀,x∈R. Moreover g(x) is positive and f(x) is non negative and ∫0kf(g(x))dx=1−e−k2, then the value of ln(gf(3)gf(2)) is _________

x2f'(g(x))g'(x)f(g(x))=(1−2x2)g'f(x).f'(x)g(f(x)) x2ddx(lnf(g(x)))=(1−2x2)ddx(lngf(x)) ∫0kf(g)dx=1−e−k2 f(g(k))=2k e−k2 ln f(g(x))=ln(2xe−x2) ddxln(f g(x))=2e−x2−4x2e−x22xe−x2=2e−x2(−2x2)+12e−x2.x=1x−2x Now, x2ddxln(fg(x))=(1−2x2)ddxln(g f(x)) ⇒ x2.1−2x2x=(1−2x2)ddxln(g f(x)) ⇒ ln(g f(x))=x2x+c ⇒ ln(g f(3)g f(2))=92−2=52

Let f(x) and g(x) are differentiable function of x such that x2(gf(x)).(f'g(x)).(g'(x))=(1−2x2)(g'f(x)).(f'(x))f(g(x))∀,x∈R. Moreover g(x) is positive and f(x) is non negative and ∫0kf(g(x))dx=1−e−k2, then the value of ln(gf(3)gf(2)) is _________

x2f'(g(x))g'(x)f(g(x))=(1−2x2)g'f(x).f'(x)g(f(x)) x2ddx(lnf(g(x)))=(1−2x2)ddx(lngf(x)) ∫0kf(g)dx=1−e−k2 f(g(k))=2k e−k2 ln f(g(x))=ln(2xe−x2) ddxln(f g(x))=2e−x2−4x2e−x22xe−x2=2e−x2(−2x2)+12e−x2.x=1x−2x Now, x2ddxln(fg(x))=(1−2x2)ddxln(g f(x)) ⇒ x2.1−2x2x=(1−2x2)ddxln(g f(x)) ⇒ ln(g f(x))=x2x+c ⇒ ln(g f(3)g f(2))=92−2=52

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Let $f (x)$ and $g (x)$ are differentiable function of x such that $x^{2} (g f (x)) . (f' g (x)) . (g' (x)) = (1 - 2 x^{2}) (g' f (x)) . (f' (x)) f (g (x)) \forall, x \in R .$
Moreover $g (x)$ is positive and $f (x)$ is non negative and $\int_{0}^{k} f (g (x)) d x = 1 - e^{- k^{2}},$ then the value of $\ln (\frac{g f (3)}{g f (2)})$ is _________

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Detailed Solution

$x^{2} \frac{f' (g (x)) g' (x)}{f (g (x))} = (1 - 2 x^{2}) \frac{g' f (x) . f' (x)}{g (f (x))} x^{2} \frac{d}{d x} (\ln f (g (x))) = (1 - 2 x^{2}) \frac{d}{d x} (\ln g f (x)) \int_{0}^{k} f (g) d x = 1 - e^{- k^{2}} f (g (k)) = 2 k e^{- k^{2}} \ln f (g (x)) = \ln (2 x e^{- x^{2}}) \frac{d}{d x} \ln (f g (x)) = \frac{2 e^{- x^{2}} - 4 x^{2} e^{- x^{2}}}{2 x e^{- x^{2}}} = \frac{2 e^{- x^{2}} (- 2 x^{2}) + 1}{2 e^{- x^{2}} . x} = \frac{1}{x} - 2 x N o w, x^{2} \frac{d}{d x} \ln (f g (x)) = (1 - 2 x^{2}) \frac{d}{d x} \ln (g f (x)) \Rightarrow x^{2} . \frac{1 - 2 x^{2}}{x} = (1 - 2 x^{2}) \frac{d}{d x} \ln (g f (x)) \Rightarrow \ln (g f (x)) = \frac{x^{2}}{x} + c \Rightarrow \ln (\frac{g f (3)}{g f (2)}) = \frac{9}{2} - 2 = \frac{5}{2}$