Critical points M = { (x,y,z) ∈ ∈ |R3 | ︎x2 + y2 + (z2/4) = 357, x + 4y = z }.

Q1. – “Critical points” M := { (x,y,z) ∈ ∈ |R3 | ︎x2 + y2 + (z2/4) = 357, x + 4y = z }.

a) Sketch the set M and explain in a few sentences

b) Explain, why the set M contains points with minimal distance from the coordinate origin.

c) Explain, why the points with minimal distance from the coordinate origin are exactly those points, for whom the square of the distance from the coordinate origin are minimal.

d) Find the points in M with minimal distance from the coordinate origin

Advice: It is easier, to consider the SQUARE of the difference

Q2. – “Classification of critical points”

Consider f: |R3 → |R a twice differentiable function, and M = { x ∈ |R3 | g ( x ) = 0 }

A regular two-dimensional manifold with g ∈ C1(|R3, |R). The function

Has the critical points ∈ M and

Find out if

Has local extrema in the critical points

And if so whether those are local minima or maxima.

The post “Critical points” M := { (x,y,z) ∈ ∈ |R3 | ︎x2 + y2 + (z2/4) = 357, x + 4y = z }..