Find a function so that in $(0,0)$ is an isolated local maximum, in $(-17,0)$ and $(0,3)$ are an isolated minimum.

Question

My task is to find a continous differentiable function by knowing located isolated extreme in point $(0,0)$ is an isolated local maximum, in point $(-17,0)$ and $(0,3)$ are an isolated minimum.

Do you know a function? I can do the proof but I can't find a function can you please help me?

Answer 1

Here is a solution. Consider the "eggbox"-shaped surface (see here) with equation :

$$z=\cos(\pi x)\cos(\pi y)\tag{1}$$

This surface has a local maximum in $(0,0)$ and, nearby, local minima in $(1,0)$ and $(0,1)$ (of course, due to periodicity, there will be an infinite number of those...)

Now, make a change of variable using the linear transformation mapping $(1,0),(0,1)$ onto $(-17,0),(0,3)$ resp., yielding the following equation of the "stretched" surface :

$$z=\cos(\pi x/17)\cos(\pi y/3)\tag{2}$$

One can easily check the properties of local extrema because expression (2) takes its values in $[-1,1]$, with value $-1$ (resp. $+1$) being taken necessarily at a local minima (resp. local maxima).