Can someone solve these three problems with clear explanation please ASAP? Find

Question

Can someone solve these three problems with clear explanation please ASAP?

Find the absolute maxima and the absolute minima of f(x, y) = x2 - y2 + 3x, whose domain is bounded by the circle of radius 2 centered at the origin. A submarine captain wants to avoid detection by an enemy destroyer. He can do this by heading toward the coldest water. If the gradient of the temperature function of the ocean is T = 2z2 i^ + ey z j^ + (4xz + ey )k^, and the sub is at the point(2, 0, l) what direction should he head to get to the coldest water the quickest? Find a point on the surface z = 2x2 + 3y2 where the tangent plane is parallel to the plane 8x - 3y - z = 0. (The point is an ordered triple.)

Explanation / Answer

For the third problem,any plane parallel to 8x-3y-z=0 can be taken as 8x-3y-z+k=0

So

z=8x-3y+k

PUt it in z=2x^2 +3y^2

you get

2(x^2 -8x)+3(y^2 +y)-k=0

Which is same as

2(x-4)^2 +3(y-0.5)^2 -134/4 -k=0

If (134/4 -k) is not equal to zero this eqn has more than soln.So it has to zero(since tangent touches at only one point).This gives K=-131/4

And the solution is x=4,y=0.5,

For 'z' use eqn z=8x-3y+k.

You get z=-73/4

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