Problem 12.2. Find all values of $a$ such that the maximum of the function $f(x)=\frac{a x-1}{x^{4}-x^{2}+1}$ is equal to 1 .

Oleg Mushkarov, Nikolai Nikolov
12.2. Since the denominator of the function is positive, the given condition means that $a x-1 \leq x^{4}-x^{2}+1$ for any $x$, and that the equality is attained for some $x$.

Let $a \geq 0$. Then $a x \leq 0$ for $x \leq 0$ and hence $a$ is the minimum of the function $g(x)=\frac{x^{4}-x^{2}+2}{x}$ for $x>0$. Since $g^{\prime}(x)=\frac{\left(3 x^{2}+2\right)(x+1)(x-1)}{x^{2}}$, it follows that this minimum is equal to $g(1)=2$. The case $a \leq 0$ can be reduced to the previous one replacing $a$ by $-a$ and $x$ by $-x$. So $a= \pm 2$.