Local minimum of a polynomial

Question

Let $a\ge 0, b\ge 0$ and $c\le 0$ We consider the polynomial $P(x)=ax_1+bx_2^2+cx_3^2$ with variables $x=(x_1,x_2,x_3)\in \mathbb{R}^3$ Is the following equivalence true?

$P$ does not have any local minimum if and only if $a-c\not=0$

Answer 1

It is very easy to determine where $P$ has a local minimum, since the contributions of each variable are separate. If $a\neq 0$, then $P(x)$ can be increased by increasing $x_1$, so $P$ has no local minimum unless $a=0$. Similarly, if $c\neq 0$, then $P(x)$ can always be increased by increasing the absolute value of $x_3$, so $P$ has no local minimum unless $c=0$. Finally, if $a=c=0$, we have $P(x)=bx_2^2$, which is minimized everywhere if $b=0$ and is minimized when $x_2=0$ if $b>0$.

So, $P$ has a local minimum iff $a=c=0$, which under your assumptions is equivalent to $a-c=0$.

Answer 2

A local minimum has to happen at a point where the gradient $\nabla P=\vec{0}$.

Here $\nabla P=\left<a,2bx_2,2cx_3\right>$.

you can immediately conclude that $a=0$ so $P=bx_2^2+cx_3^2$ which iseither a quadric surface when projected above the $x_2x_3$-plane or degenerate with $P=bx_2^2$ or $P=cx_3^2$. Which gives 3 cases that have local minimums:

• $b\neq 0$, $c\neq 0$ the $P$ is a quadric surace and is either a hyperbolic paraboloid if $bc<0$ which has no local minimums or a elliptical paraboloid when $bc>0$ it only has local minimums when $b>0$ and $c>0$ at the point $(x_1,0,0)$ since we're free to set $x_1$ to whatever we like.
• $b=0$,$c\neq 0$ then $P=cx_3^2$ is a parabola and has local min at $x_3=0$ when $c>0$ only so the points $(x_1,0,0)$ are local mins
• $c=0$,$b\neq 0$ then $P=bx_2^2$ is a parabola and has local min at $x_2=0$ when $b>0$ only so the points $(x_1,0,0)$ are local mins
• $a=b=c=0$ then $P=0$ and every point is a local minimum (quite trivially). Though some authors may try to exclude constants here, depends I guess.