Lower bond for the natural exponential function with quadratic argument

Question

Consider the following natural exponential function:

$f\left( {\bf{x}} \right) = {e^{{{\bf{a}}^T}{\bf{x}} + \frac{1}{2}{{\bf{x}}^T}{\bf{Cx}}}}$

where $\bf C$ is a positive definite matrix. Is there any tight lower bound for this function?

Answer 1

Guide:

Notice that the exponential function is an increasing function, hence the lower bound is the smallest when $g(x)=a^Tx+\frac12x^TCx$ attains the smallest value.

You can differentiate $g(x)$ to solve for the minimal value.