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I have function $f(x,y)=\sin(xy)$ in two variables. I found critical point $(0,0)$.

$f(x,y)$ has saddle point at $(0,0)$, because $D=f_{xx}(0,0)f_{yy}(0,0)-f_{xy}(0,0)<0$

So, I got saddle point, but what about other critical points? Please help me. Thanks in advance.

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2 Answers 2

You can find the other points calculating the gradient of your function f and making it equal to 0. Then solve the equation system and you'll get all the critical points. Using the second derivate criteria you can determine whether they're max, mins or saddles.

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Hint: Well, notice $\cos(xy) = 0$ implies $xy = \pi \frac{2n+1}{2} $ for $n \in \mathbb{Z}$. Now you should be able to find the other critical points!

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i got $\cos(xy)=0$ it means $xy=(2n\pm 1) \frac{\pi}{2}, n\in \mathbb{z}$ but i cant get value of D in integer at this point –  Siddhant Trivedi Nov 3 '12 at 7:04
    
just edited.... –  Chasky Nov 3 '12 at 7:06

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