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.

Do you mean functions like $f(x) = e^x$? That's a non-negative function, but it doesn't have a smallest value. — Matti P., Mar 31 '21 at 12:02
Can you clarify a bit more? Are you looking to see whether other extrema than minima are possible with such functions, or do you want functions that only consist of minimal points? — S.Farr, Mar 31 '21 at 12:03

score 0 · Answer 1 · answered Mar 31 '21 at 12:05

0

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

answered Mar 31 '21 at 12:05

TomKern

score 0 · Accepted Answer · answered Mar 31 '21 at 12:21

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).
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.

2 Answers2