How to calculate the absolute extrema for this equation Basically, I am trying to find the absolute extrema for this problem, but I am stuck after a few steps.
$$y=\frac{4}{x}+\tan\left(\frac{\pi x}{8}\right),\quad x\in[1,2].$$
I have figured out the derivative, which is
$$y'=-\frac{4}{x^2}+\sec^2\left(\frac{\pi x}{8}\right)\cdot\frac{\pi}{8}.$$
However, after reaching this step, I am stuck. I have tried setting the derivative equal to zero, but I do not get anywhere as I don't know what to do next. How can I solve this problem? Any help is appreciated. Thanks.

To continue after your last step . Simplifying after setting the derivative to zero,
$$ \dfrac{x}{2}=\cos \sqrt{\dfrac{\pi x}{8}}$$
Can be solved by iterative methods numerically, but no closed form.
