<s>[INST] The minimum and the maximum of $y=\sin^2x/(1+\cos^2x)$ I was asked to find the minimum and maximum values ​​of the functions:

  
*
  
*$y=\sin^2x/(1+\cos^2x)$;
  
*$y=\sin^2x-\cos^4x$.
  

What I did so far:


*

*$y' = 2\sin(2x)/(1+\cos^2x)^2$
How do I check if they are suspicious extrema points? After this function is cyclical and therefore only section that is not $(-\infty,\infty)$ can there be a local minimum/maximum.

*$y' = \sin(2x)+4\cos^3(x)\cdot\sin(x)$
Any suggestions?
 [/INST] If  you don't really need to use derivatives,
$1.$
Clearly, $y=\frac{\sin^2x}{1+\cos^2x}\ge 0$ in fact $=0$ if $\sin x=0$
$y-1=\frac{\sin^2x}{1+\cos^2x}-1=\frac{\sin^2x-(1+\cos^2x)}{1+\cos^2x}=-2\frac{\cos^2x}{1+\cos^2x}\le 0$ in fact $=0$ if $\cos x=0$
$\implies y-1\le 0\iff y\le 1\implies 0\le y\le 1$
$2.$  $$y=\sin^2x-\cos^4x=1-\cos^2x-\cos^4x=1-\cos^2x(1+\cos^2x)$$
$$\text{Now, }0\le \cos^2x\le1\implies 1\le\cos^2x+1\le2 \implies 0\le  \cos^2x(\cos^2x+1)\le2$$
$$\implies 0\ge-\cos^2x(\cos^2x+1)\ge-2\iff 1\ge1-\cos^2x(\cos^2x+1)\ge-1$$
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