2014
Problem - 2958
Let $f(x)=x^{-\frac{k^2}{2}+\frac{3}{2}k+1}$ be an odd function where $k$ is an integer. If $f(x)$ is monotonically increasing when $x\in(0,+\infty)$, find all the possible values of $k$.
Because $f(x)$ monotonically increases, we find $-\frac{k^2}{2}+\frac{3}{2}k+1 > 0 \implies -1 < k < 4 \implies k = 0, 1, 2, 3$
Because $f(x)$ is an odd function, we conclude $k$ can only take $\boxed{1}$ or $\boxed{2}$.