Good morning,
I stumbled myself onto the following apparently deceiving problems. Since I didn't make any reasonable progress I ask for ideas or hints.
The problem is very simple: suppose that we have the following two functions $$\phi(x)=\frac{1}{\sin(x)^2}-\frac{\cot(x)}{x},\quad \psi(x)=\frac{x^2}{\sin(x)^2}+x\cot(x).$$ Since the functions are even, we restrict to the positive real line.
The question is, for any fixed value of $c$, find the minimum of $\psi(x)$, among all the solutions of $\phi(x)=c,$ if there are any, so let's say for $c\geq2/3$.
The reasonable guess is that the minimum is achieved iff $x\in[0,\pi)$, but I'm really stucked.
Also, it may be worth notice that $\psi(x)'=x^2\phi(x)'$.
Thanks in advance,
Guido