1st order condition differentiable f with convex domain is convex IFF:
\begin{equation} f\left(y\right) \geq f\left(x\right) + \nabla f\left(x\right)^{T} \left(y-x\right), \forall x,y \in \text{dom } f \end{equation}
“the function is everywhere above the Taylor approximation” => “first-order Taylor approximation of f is a global underestimator of f.” 2nd order condition for twice differentiable f with convex domain, we have: f is convex IFF \nabla^{2} f\left(x\right) \succeq 0, \forall x \in \text{dom } f (i.e. that the Hessian is PSD) if \nabla^{2} f\left(x\right) \succ 0 \forall x \in \text{dom } f, then we call f strictly convex (i.e. that the Hessian is PD) you may enjoy using Cauchy-Schwartz Inequality to show these. We also call the PSD-ness of \nabla^{2}f\left(x\right) the “curvature.”