# Maximization of a differentiable and nonlinear function over a bounded space

I have a nonlinear bi-variate optimization problem like $$\max \: f(x,y)$$ where $$f(x,y)$$ is a nonlinear and differentiable function of both variables, and $$0\le x\le 1$$, $$\:0\le y \le ub$$. In order to find the optimal solution, can I find the stationary points, $$\nabla f(x,y)=0$$, and boundary points and evaluate the objective function at the corners, such as $$(0,0), (0,ub), (1,0),(1,ub)$$ points and stationary points $$\nabla f(x,y)=0$$ (KKT points)? The highest order in the function is 3 and in form of $$xy^2$$ or $$yx^2$$?

• Is the function $f$ concave?
– joni
Jul 14 at 7:34
• Stationary points (gradient = 0) and KKT points are two different things. Which do you mean? Jul 14 at 17:53
• Yes, you are right. I mean aggregation of KKT points and 4 vertices of the feasible space. The function is not concave, and the objective function it is bounded Jul 15 at 1:27

• +1, but actually for KKT to be necessary, the objective function $f$ must be continuously differentiable (which it would be if $f$ is a polynomial, as the OP might be suggesting) it is), not just differentiable as stated in the question. Jul 15 at 20:02