Problem with Lagrange multiplier (max and min distance from a cone)

Question

I need to find the points of $C=\{(x,y,z)\in \mathbb R^3:4\sqrt{x^2+y^2}\le z\le 1 \}$ with max and min distance from $(-1,0,3)$ I think the solution is:

• Looking for the distance with Lagrange multiplier on the upper surface of the cone
• Looking for the distance with Lagrange multiplier on the external surface of the cone

But i don't know how to parameterize those sets (the upper surface and the external surface).

Can someone help me?

Answer 1

Hint:

What is the set $4\sqrt{x^2+y^2}\leq 1, z=1$?
What is the set $4\sqrt{x^2+y^2}=z, 0 \leq z\leq 1$?

If you need to help to visualize this geometrically, notice that $\sqrt{x^2+y^2}$ is the distance from $(x,y,z)$ to the $z$-axis, or the radius in polar $3$D coordinates.