# Find maximum and minimum value of function of three variables on the set $E$

$$f(x,y,z)=4x+2y+z$$ $$E=\{(x,y,z) \in R : (x+1)^2+4y^2+4z^2=4\}$$

I know I should write here what I already did but I could come up with literally nothing. Should I just find extreme values of $g(x,y,z):=4x+2y+z-((x+1)^2+4y^2+4z^2-4)$ does it even make sense?

• Seems like this problem is very well suited for Lagrange multipliers. Do you know that method? – Matthew Leingang Nov 5 '17 at 18:11
• I haven't heard anything about that, so no I don't – UfmdFkiF Nov 5 '17 at 18:16
• Here you go, then: Calculus III \- Lagrange Multipliers – Matthew Leingang Nov 5 '17 at 18:17
• I see but if the question was "find the maximum value of $f$ on $E$" would there be some method without using Lagrange multipliers? – UfmdFkiF Nov 5 '17 at 18:26

Hint (without Lagrange multipliers)

Write $x+1 = 2 \cos\theta$ and $y = \sin\theta \cos\varphi$ and $z=\sin\theta\sin\varphi$. You are left with finding the extrema of a smooth periodic function in $\mathbb{R}^2$.