Why is $f (x) = x' Q x + c x + b$ convex if $Q$ is PSD?

Question

The function $f(x)=x^TQx +c^Tx +d$ is convex if $Q$ is positive semidefinite ( $x^TQx \ge 0 \forall x\in \mathbb{R} $).

How to prove this statement?

Can $Q$ not be positive semidefinite and $f(x)$ be convex?

Answer 1

Second Fréchet derivative is given by $$f^{\prime \prime}(x).(h,k) = h^T Q k + k^T Q h$$

As $Q$ is supposed to be PSD, $f$ is convex.