Find number of continuous functions satisfying the equation $4\int_{0}^{\frac{3}{2}}f(x)dx+125\int_{0}^{\frac{3}{2}}\frac{dx}{\sqrt{f(x)+x^2}}=108$

Question

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

(A)$0$

(B)$1$

(C)$2$

(D)greater than $2$

My Attempt:

Given equation can be written as $$4\int_{0}^{\frac{3}{2}}(f(x)+x^2)dx+125\int_{0}^{\frac{3}{2}}\frac{dx}{\sqrt{f(x)+x^2}}=\frac{225}{2}$$ Now I am trying to use some inequality like AM-GM or CS so that a unique function happens so that inequality is valid only when equality holds but am not able to get it.

Answer 1

Let $g(x)=f(x)+x^2$ first of all. Also note that $g(x)\geq 0$. Now, $$4\int_0^{3/2}g(x)\textrm dx+125\int_0^{3/2}\dfrac{1}{2\sqrt{g(x)}}\textrm dx+125\int_0^{3/2}\dfrac{1}{2\sqrt{g(x)}}\textrm dx$$ $$\geq3\int_0^{3/2}\left(4g(x)\cdot\dfrac{125}{2\sqrt{g(x)}}\cdot\dfrac{125}{2\sqrt{g(x)}}\right)^{1/3}\textrm dx$$ Hope this helps. BTW I also took the same exam. ;)