I have a question about partial differential , stationary points and to classify those.
$$f(x,y) = e^x + x + \frac{xy^2}{2} - 2xy +6y - \frac {3y²}{2}$$
so the partial diffs are(quite sure it's right):
$$f_y = xy-2x+6-3y = (x-3)(y-2)$$ $$f_x = e^x +1 + \frac{y}{2}(y-4)$$ $$f_{yy} = (x-3)$$ $$f_{xx} = e^x $$
My task was to show that $(0,2)$ is a stationary point and then i should classify the point. I've shown it by setting $f_x,f_y = 0$. What i don't manage to do is classifying the point. Can i get some help with that? Thanks
You have to look at the Hessian matrix in the given point and look for its eigenvalus (or at least its sign).
If you have a 2x2 matrix $H$ (as in your case) it is very easy to find the sign of the eigenvalues since: $$ \det H = \lambda_1 \lambda_2\\ \mathrm{tr} H = \lambda_1 + \lambda_2 $$ so if $\det H < 0$ then you have a saddle point, otherwise you have to check the trace: if it is positive you have a local minimum if it is negative you have a local maximum.