<s>[INST] How to find the maximum and minimum values of $\frac{8x(x^2-1)}{(x^2+1)^2}$ algebraically? The function is $f(x) = \frac{8x(x^2-1)}{(x^2+1)^2}$.
I have tried using calculus, only to fail.
 [/INST] Suppose $b$ is in the range of the given function. Then equation
$$ \frac{8x(x^2-1)}{(x^2+1)^2} = b $$
$$ \frac{8x^2(x-{1\over x})}{x^2(x+{1\over x})^2} = b $$
Mark $t =  x-{1\over x}$ and $t$ takes all real values. Then we have 
$$ b= \frac{8t}{t^2+4} $$ If $t$ is positive then $b\leq 2$ since $4t\leq t^2+4$ or $(t-2)^2\geq 0$ is true. If $t$ is negative then $b\geq -2$. So the range is $[-2,2]$ since $b(t)$ is continious function. 
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