# How to solve this constrained maximization problem?

Does anyone have an idea of how to tackle the following maximization problem?

Maximize the function $$f(x,y,z) = x - y - \alpha z^2$$, $$\alpha > 0$$, under the following constraints:

• C1: $$x>0$$, $$\:\: y>0$$, $$\:\: 0
• C2: $$y - \frac{1}{2} x^2 + c_1x \geq 0$$, $$\:\: c_1>0$$
• C3: $$y - \frac{1}{2} x^2 + \frac{1}{2} (c_2 - A z )^2 < 0$$, $$\:\: c_2,A>0$$

I would be grateful for any help/suggestions! :)

• What is $\alpha$ ? – Ken Mar 1 at 10:56
• If you keep all the inequality strict, then there is no maximum clearly. $f$ does not have any critical points. – Pebeto Mar 1 at 15:45
• @Pebeto Thank you! – Apollo13 Mar 2 at 12:03