Question

Find the global maximum and minimum values of
f(x, y) = x^3 − y^2 + xy on the closed triangular region with vertices (0, 0), (1, 0), (0, 1).

Answer 1

The global maximum is 1 and the global minimum is 1.
Let me guide you through it.
The gradient of f is
\[
\left\{3 x^2+y,x-2 y\right\}

\]
Find first where the gradient is zero at the interior of your domain.
Than examine what happened at the boundary.
x=0
y=0
y =-x +1