How to show that the function $\log(1+\alpha t^2)$ is quasi convex?

Question

I need to show that the function $f(t)=\log(1+\alpha t^2)$ is quasi convex. I checked its quasi convexity by plotting it using matlab as follows.

I really appreciate if you could please give me some directions to prove its quasi convexity.

Answer 1

Quasiconvexity of $f$ means that for all $a\in\mathbb R$, the set of all $t\in \mathbb R$ such that $f(t)\leq a$ is an interval. Can you compute that set ?