Let $f:\mathbb R\to\mathbb R$, maps a point $x \in \mathbb R$. $f$ is twice differentiable. Show that if second derivative is positive for all $x$ then $f$ is convex
Is there anyway to prove this without using MVT/definitions of derivatives?
Basically can this be proven with just the definition of convexity/concavity and maybe quasi-convexity and lower contours. The definition we learned use for convexity is: $f$ is convex if $$ f(\theta x_a + (1-\theta) x_b) \leq \theta f(x_a) + (1-\theta) f(x_b). $$