If $x>0$, then find the greatest value of the expression $ \frac{x^{100}}{1+x+x^2+x^3+\cdots+x^{200}}$ 
If $x>0$, then find the greatest value of the expression $ \dfrac{x^{100}}{1+x+x^2+x^3+\cdots+x^{200}}$     

This expression simplifies to $ \frac{(x^{100})(x-1)}{x^{200}-1}$ using sum of n terms of GP. Now one can find the maxima by equating the derivative to zero. But is there any other way not involving calculus to get the maximum value?

Hint: write it as
$$
\begin{align}
& \quad \frac{1}{\cfrac{1}{x^{100}} + \cfrac{1}{x^{99}} + \cdots + \cfrac{1}{x} + 1 + x + \cdots + x^{99} + x^{100}} \\
&= \frac{1}{1 + \left(x+\cfrac{1}{x}\right) + \left(x^2+\cfrac{1}{x^2}\right) + \cdots +  \left(x^{100}+\cfrac{1}{x^{100}}\right)}
\end{align}
$$
and use the fact that $a+\frac{1}{a} \ge 2$ for $a \gt 0$, with equality iff $a=1$.
