Find the points of maximum and minimum of the function $$f(x,y,z) = 2x + y - z^2$$ in the compact space $$C = \{(x,y,z) \in \mathbb{R}^3 : 4x^2 + y^2 -z^2 = -1,z\ge 0, 2z \le 2x + y + 4\}$$
So, I have the answer for this, which is:
Minimum: $$ -19 - 6\sqrt{7} $$ Maximum: $$ -\frac 12 $$
I already found the minimum value, but I'm having trouble to find the maximum. Can someone help me out?
hint:
$t=2x+y,z^2-1=4x^2+y^2 \ge \dfrac{t^2}{2} \implies f\le t-\dfrac{t^2}{2} -1$
$z\le \dfrac{t+4}{2} \implies \left(\dfrac{t+4}{2} \right)^2-1 \ge\dfrac{t^2}{2} $