Math Is Fun Forum

dexza666

Member

Registered: 2008-04-21

Posts: 5

extrema

find all critical points of
f(x, y)=e^x(1-cos y)

and classify these critical points.

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: extrema

I have also never covered this

Heres a plot of the function, y running roughly from right to left, x running roughly from front to back, f(x,y)=z running roughly bottom to top. I've plotted a small section of it, plotting a larger section this pattern repeats and repeats, and by zooming in like this, and even further, you can see that the lines y = 2kπ are sort of speak 'critical' lines backing up my work above

I'll now start on trying the next part of classifying them, although already from the graph you can infer i think.

From the graph, and or equations below all the critical points of the function have positive second partial derivitive in y, and zero second partial derivitive in x, and zero second partial derivitive in x and y. what kind of classification is given to that?

Last edited by luca-deltodesco (2008-04-22 19:10:18)

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luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: extrema

A quick google gives me the second deritivite test for functions of two variables, applying it here

so the second derivitive test is inconclusive >.>

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luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: extrema

going down another route.

the hessian matrix is given by

which is singular, so can't use this method either >.>

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