Boundedness of quadratic forms

Question

How can I prove that a function in quadratic form $F(x)=1/2 (x^TAx) + b^T x + c$ is either bounded below or above?

What does it imply in terms of convexity of the function?

Many thanks in advance.

Answer 1

Consider the matrix \begin{align} A = \begin{pmatrix} 1 & 0 \\ 0 & - 1 \end{pmatrix} \end{align} and $b = 0$ vector and $c = 0$, then we have the function \begin{align} F(x, y) = \frac{1}{2} (x, y) \begin{pmatrix} 1 & 0 \\ 0 & - 1 \end{pmatrix} \begin{pmatrix} x\\ y \end{pmatrix} = \frac{1}{2}( x^2-y^2) \end{align} which is neither bounded above nor below.