Finding Maximum and Minimum of Curve of Intersection

Question

I need help with part b) of this question, I don't understand what they did. I understand what the question is asking, but why is the plane equation being set to y rather than x? What method should be used for this problem?

Answer 1

Since on the surface it is obvious that z ≤ 1, and since (0,0,1) clearly lies on both sets, the maximum is z=1. Then by symmetry, it is obvious that the minimum occurs when x=y, which quickly gives x = (1-z)/2. My suggestion here is that looking at the problem is sometimes more effective than puzzling out the book's solution.

However, the book simply eliminated one variable to reduce to the case of a curve in the (x,z) plane. you could just as well have eliminated x, getting a curve in the (y,z) plane. Their next step used the fact that at the point of a curve g(x,z) = constant in the (x,z) plane where z is maximal, the derivative ∂g/∂x should be zero, i.e. at such a point the tangent line should be parallel to the x axis, so moving in the x direction is moving in a direction along which the value of g is almost constant.