This was an exam question I found and I need some help figuring it out
A village is founded in a mountain area of which the height (in meters) is calculated by $$H(x,y)=500e^{-(x-2)^2-(y-x)^2}$$ a) Check if $H$ has a local maximum. Is this the global maximum? Where is this achieved and what is the height?
b) The centre of the village is found at point $(0,0)$. From there a road goes around the village at a constant height. On which point does it have the highest $y$-value and on which the lowest?
For a) I started with normal techniques to calculate the local max and found $(2,2)$ to be it. But i'm not quite sure how to get b).
Hint
The equation of whole the village would be$$e^{-4}=e^{-(x-2)^2-(x-y)^2}$$therefore $$(x-2)^2+(x-y)^2=4$$By defining $$x-2=2\cos\theta\\y-x=2\sin \theta$$we obtain $$x=2+2\cos\theta\\y=2+2\sin \theta+2\cos\theta$$from which the maximum and minimum could be easily found.