I have this function ($x>0$): $$ f(x)=\left(x^2+2\right)^2 \cosh (x)-\left(2-x^2\right)^2 \cosh (3 x)-\left(\frac{x}{2}\right)^2 $$ Here is the picture of the function and its first derivative
I want to prove that this function crosses the $x$ axis only twice. Any hints or suggestions are welcome.
Evaluate $f(0.8),f(1)$ and $f(2)$. Then note that the function is continuous, and since $f(0.8) < 0 < f(1)$ and $f(1) > 0 > f(2)$, by the IVT there must be 2 roots.
As is pointed out by @user247327 in a comment below, this demonstrates there must be at least two roots. Not clear if you want to prove there are exactly 2 roots.