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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

Emanuele Paolini
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kirkegaard
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1 Answers1

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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.

Emanuele Paolini
  • 21,447
  • There are no other way? can't remember we have been through that method in class. But i cant remember any other way either so.. – kirkegaard Oct 21 '15 at 10:13