Particle projected from above a dome

[Steve4Physics]

It applies to any function of the form $af(x) + \frac b {f(x)}$ , which that one is.

Providing $af(x)$ and $\frac b {f(x)}$ are both positive in the domain of interest!

 

[PeroK]

Providing $af(x)$ and $\frac b {f(x)}$ are both positive in the domain of interest!

The main criterion is that their ranges overlap. I forgot to add that above.

 

[TSny]

The main criterion is that their ranges overlap. I forgot to add that above.

Yes, in my example I should have said that the first term decreases from $\infty$ to $0$ while the second term increases from $0$ to $\infty$ for $0<x<\pi/2$. Thus, there is an $x$ in this domain where the two terms are equal.

 

[Bling Fizikst]

In general $(a,b>0)$ , if $z>0$ then $$az+\frac{b}{z}\geq 2\sqrt{az\cdot\frac{b}{z}}=2\sqrt{ab}$$ If $z<0\implies -z>0$ then $$-az-\frac{b}{z}\geq 2\sqrt{-az\cdot\frac{-b}{z}} =2\sqrt{ab}\implies az+\frac{b}{z}\leq -2\sqrt{ab}$$