Given a function

$$f(x) = \frac{1}{2}x^TAx + b^Tx + c$$

, where $A \succeq 0$, $A \in \mathbf S^n$, and equation $\nabla f = Ax + b = 0$ has no solution. We need to prove that $f(x)$ is unbounded below.

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Here's my approach:

$Ax + b = 0$ has no solution means $b \notin \mathcal R(A)$.

$$b \notin \mathcal R(A) \Rightarrow \mathcal N(A) = \mathcal N(A^T) \ne \{0\}$$

so we have:

$$\exists\ v \ne 0,\ Av = 0$$

Let $x = tv$, where $t \in \mathbb R$, we have

$$f = tb^Tv + c$$

This is a linear function of $t$, and $f$ is unbounded below if $b^Tv \ne 0$. The problem is I can't think of a way to prove $b^Tv \ne 0$.