<s>[INST] Bound of a function $f_n=\frac{x^2}{x^2+(1-nx)^2}$ Let $\displaystyle f_n=\frac{x^2}{x^2+(1-nx)^2}$ where ($0\le x\le 1,\; n=1,2,3,...$)  
Then $|f_n(x)| \le M$.
Find this $M$.  
The answer is $1$.
Without any restriction of n, how can we find that bound?
 [/INST] Let $\frac{x^2}{x^2+(1-nx)^2}=y$
or $x^2(y+yn^2-1)-2nyx+y=0$
As $x$ is real, $(2ny)^2\ge 4y(y+yn^2-1)$
$\implies y^2-y\le 0\implies y(y-1)\le 0\implies 0\le y\le 1$
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