<s>[INST] Global extrema for $\frac{(q^k-1)(q^{k+1}-2q^k+1)}{{q^k}(q-1)(q^{k+1}-1)}$? Does the function
$$\frac{(q^k-1)(q^{k+1}-2q^k+1)}{{q^k}(q-1)(q^{k+1}-1)}$$
have either a global minimum or a global maximum for $q \geq 5$ and $k \geq 1$?
WolframAlpha is unable to find any.
WolframAlpha computation for the global minimum
WolframAlpha computation for the global maximum
 [/INST] 
can we do better than this?

No, we cannot.
Let $f(k)$ be your function.
Then, we have
$$f'(k)=\frac{ (q-4)q^{2 k + 1} +2 q^{k + 1}  + 2 q^{2 k} - 1}{q^k(q - 1) (q^{k + 1} - 1)^2}\ln q$$
which is positive for $k\ge 1$ and $q\ge 5$.
So, we see that $f(k)$ is increasing for $k\ge 1$.
Since
$$f(1)=\frac{q-1}{{q}(q+1)},\qquad \lim_{k\to\infty}f(k)=\lim_{k\to\infty}\frac{(1-\frac{1}{q^k})(1-\frac{2}{q}+\frac{1}{q^{k+1}})}{(q-1)(1-\frac{1}{q^{k+1}})}=\frac{q-2}{q(q-1)}$$
we have
$$\frac{q-1}{{q}(q+1)}\le f(k)\lt \frac{q-2}{q(q-1)}$$

Added : 
Let $f(q)$ be your function.
We get, with some help of WA,
$$f'(q)=\frac{- (q^{k + 1} - k q + k - q) (q^{2k+1}(q-4)+2q^{k+1}+2q^{2k}-1)}{q^{k+1}(q-1)^2(q^{k+1}-1)^2}$$
Here, let
$$g(q):=q^{k + 1} - k q + k - q$$Then,$$g'(q)=(q^k-1)(k+1)\gt 0$$So, $g(q)$ is increasing and $$g(q)\ge g(5)=5^{k + 1} - 4k - 5\gt 0$$
It follows that $f(q)$ is strictly decreasing for $q\ge 5$.
Since $\displaystyle\lim_{q\to\infty}f(q)=0$, we have
$$0\lt f(q)\le f(5)=\frac{(5^k-1)(5^{k+1}-2\cdot 5^k+1)}{4\cdot 5^k(5^{k+1}-1)}\lt \frac{3}{20}$$
</s>