Let \(A\) be a positive definite \(n \times n\) real matrix, \(b \in \mathbb{R}^{n}\), and consider the quadratic polynomial

\[

Q(x):=\frac{1}{2}\langle x, A x\rangle-\langle b, x\rangle

\]

a) Show that \(Q\) is bounded below, that is, there is a constant \(m\) so that \(Q(x) \geq m\) for all \(x \in \mathbb{R}^{n}\).

b) Show that \(Q\) blows up at infinity by showing that there are positive constants \(R\) and \(c\) so that if \(\|x\| \geq R\), then \(Q(x) \geq c\|x\|^{2}\).

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c) If \(x_{0} \in \mathbb{R}^{n}\) minimizes \(Q\), show that \(A x_{0}=b\). [Moral: One way to solve \(A x=b\) is to minimize \(Q .\) ]

d) Give an example showing that if \(A\) is only positive semi-definite, then \(Q(x)\) may not be bounded below.