Does $f_n(x)=\cos^n(x)(1-\cos^n(x))$ converge uniformly for $x$ in $[π/4 , π/2]$?
Its clear to see that the point-wise convergence is to $0$. By finding the derivative I obtained that the maximum of $f_n$ is when $\cos^n(x)=1/2$ and that $\sup |f_n(x)-0|= 1/2\cdot(1-1/2)=1/4$ which would indicate that the function does not converge uniformly, however I'm not sure the max is obtained in the given interval so I don't know if that's true.
Thanks
For all $x \in [ \pi /4, \pi /2]$ you have $0 \le \cos x \le \frac{\sqrt{2}}{2}$. For all $n$ $$|f_n(x)| = \cos^n x (1- \cos^n x) = \cos^n x - \cos^{2n} x \le \cos^n x \le \left( \frac{\sqrt{2}}{2} \right)^n \to 0$$ so you have uniform convergence.
The problem with your argument is that $$\cos^n x = \frac{1}{2} \Leftrightarrow x = \arccos 2^{-\frac{1}{n}}$$ but $$\lim_n \arccos 2^{-\frac{1}{n}} = 0 \notin [ \pi /4, \pi /2]$$ so you are going outside your domain.