Finding the local minimum and local maximum. $f(x,y) = ye^x - 3x - y +2$. If someone has time please check if i am making mistakes?
$\frac{\partial f}{\partial x} = ye^x - 3$. equation i
$\frac{\partial f}{\partial y} = e^x-1$. equation ii
Finding the stationary points, i get
$ln(e^x) = ln(1) == > x = 0$ and i get y from second equation equals to $ ye^x - 3 ==> y = 3$
stationary points $(0,3)$
$\frac{\partial^2 f}{\partial x^2} = ye^x$
$\frac{\partial^2 f}{\partial y^2} = 0$
$\frac{\partial f}{\partial x \partial y} = e^x$
$\nabla = (\frac{\partial^2 f}{\partial x^2})(\frac{\partial^2 f}{\partial y^2}) - (\frac{\partial^2 f}{\partial x \partial y})$
$\nabla = (3)(0)-(1) ==> (\nabla < 0) ==>$ a and b is saddle point?