I have some difficulties with the following problem:
$$ f:\mathbb{R}^n \to \mathbb{R} $$ $$ f(x) = \sum_{i=1}^{n} (e^{x_i} - b_ix_i) $$
$$ b_i > 0, i = 1,..., n. $$
I want to prove that :
- $f$ is continuous and coercive,
- There is at least one global minimum point.
Note: $f$ is called coercive if: $$ \lim_{\left \| x \right \| \rightarrow +\infty } f(x) = + \infty $$