<s>[INST] Maximum of $\frac{1+3a^2}{(a^2+1)^2}$ What is the maximum value of
$\displaystyle{{1 + 3a^{2} \over \left(a^{2} + 1\right)^{2}}}$, given that $a$ is a real number, and for what values of $a$ does it occur ?.
 [/INST] Let $a^2=x$.
Hence, $\frac{1+3x}{(1+x)^2}\leq\frac{9}{8}$ it's $(3x-1)^2\geq0$.
The equality occurs for $x=\frac{1}{3}$, which says that the answer is $\frac{9}{8}$.
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