Does a function ( $f(x)$ )is convex iff $f''(x)>0$?

Question

I don't know when a function is convex. My documents from different resources tell me different answers. One of them told me that a function $f(x)$ is convex if and only if $f ''(x)>0$, but another told me it must be $f''(x)<0$. I need your help, please tell me the truth. Thanks

Answer 1

Convex is for $f^{\prime\prime} \geq 0$. Remember it that way: you want the graph of the function to look like a "valley," i.e. to have a shape similar to the parabola $y=x^2$.

If the function is twice differentiable, for that, the slope of the function has to become greater and greater, i.e. the derivative $f^\prime$ has to be monotone non-decreasing. But this in turn means the second derivative $f^{\prime\prime}$ (the derivative of $f^\prime$) must be non-negative.

(For $f^{\prime\prime} \leq 0$, the function is concave. If $f^{\prime\prime} > 0$, it is strictly convex, and if $f^{\prime\prime} < 0$, it is strictly concave.)

Caveat: this is when the function is twice differentiable (and on an interval). Convexity and concavity are defined even if $f$ is not differentiable (and a fortiori twice differentiable): the definition is a bit more general, but in the case of $f$ twice differentiable they are equivalent.