The number of continuous functions f :left[0, | vedclass

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Question

(b)

Let $f(x)+x^2=g^2(x) \quad(g(x) > 0)$

$\Rightarrow \int_0^{3 / 2}\left(4\left( g ^2( x )- x ^2\right)+\frac{125}{ g ( x )}\right) dx =108

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$1$

Vedclass · Answer

(b)

Let $f(x)+x^2=g^2(x) \quad(g(x) > 0)$

$\Rightarrow \int_0^{3 / 2}\left(4\left( g ^2( x )- x ^2ight)+\frac{125}{ g ( x )}ight) dx =108$

$\Rightarrow \int_0^{3 / 2}\left(4 g ^2( x )+\frac{125}{ g ( x )}ight) dx =108+\left.\frac{4 x ^3}{3}ight|_0 ^{3 / 2}$

$=108+\frac{9}{2}=112.5$

Also $4 g^2(x)+2 \cdot \frac{125}{2 g(x)} \geq 3(125-125)^{1 / 3}$

(From A.M. $\geq$ G.M.)

$\Rightarrow 4 g^2(x)+\frac{125}{g(x)} \geq 3 	imes 25$

$\Rightarrow \int \limits_0^{3 / 2}\left(4 g ^2( x )+\frac{125}{ g ( x )}ight) dx \geq \int \limits_0^{3 / 2} 75 dx =112.5$

$\Rightarrow$ only equality holds

$\Rightarrow 4 g ^2( x )=\frac{125}{ g ( x )}$

$g ( x )=\frac{5}{2}$ or $f ( x )=\frac{25}{4}- x ^2$

The number of continuous functions $f :\left[0, \frac{3}{2}\right] \rightarrow(0, \infty)$ satisfying the equation $4 \int \limits_0^{3 / 2} f(x) d x+125 \int \limits_0^{3 / 2} \frac{d x}{\sqrt{f(x)+x^2}}=108$ is

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