Does this multivariable non-negative function only have minima points?

Question

Consider multivariable functions of the form: $$\left(\sum_{i=1}^n a_ix_i\right)^2$$ Can they only have minima points? If so, why?

I tried to plot some functions on Desmos and it looks like my hypothesis is correct, but maybe I'm missing something.

Edit: I removed the first question, as I realized it was too naïve.

Answer 1

1. No. One example is constant functions : every point is simultaneously a global maximum and a global minimum. A less degenerate example is

   having a local maximum at $x = 0$. Another is
   In fact $\mathrm{e}^{P(x)}$, where $P(x)$ is a polynomial in $x$ having a global maximum (respectively, local maximum), is an everywhere positive function having a global maximum (resp., local maximum).

2. Yes or maybe, depending on your definitions. Compute the various first partial derivatives to discover that the only possible critical point is at the origin, $x = (0,\dots, 0)$. Here, the function is $0$, so there is both a local minimum and a global minimum at the origin. However, note that if all the $a_i = 0$ then every point is simultaneously a global minimum and a global maximum.

Answer 2

The function $x^2 e^{\left(-x^2\right)} + 1$ has global maxima at $x = \pm 1$. https://www.desmos.com/calculator/llxnomd24q