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Came across this problem, but it has stumped me:

Let S denote the unit sphere x^2 + y^2 + z^2 = 1, and let

u = x + 2y + 3z

be temperature at points everywhere in 3-space.

Find the hottest and coldest points on the unit ball x^2 + y^2 + z^2 <= 1

I figured out that we need to clculate the partial derivatives in each 3 co-ordinate directions and need them all to be zero... while ensuring that we are in the unit ball. Is this just down to trial and error or is there a trick involved? Any hints?

Many Thanks,

Sam