- #1
scubakobe
- 5
- 0
1. If f(x,y)=e[itex]^{x}(1-cos(y))[/itex] find critical points and classify them as local maxima, local minima, or saddle points.
I found the partials and mixed partial for the second derivative test as follows:
f[itex]_{x}[/itex]=-e[itex]^{x}[/itex](cos(y)-1)
f[itex]_{y}[/itex]=e[itex]^{x}[/itex](sin(y))
f[itex]_{xx}[/itex]=-e[itex]^{x}[/itex](cos(y)-1)
f[itex]_{yy}[/itex]=e[itex]^{x}[/itex](cos(y))
Knowing this, and that e[itex]^{x}[/itex] does not equal 0, then the critical points are periodic at 2∏n, where n is even intervals.
However, I get inconclusive when plugging it all into the second derivative test. And a quick query in WolframAlpha shows that there are indeed critical points, however no local maxima,minima or saddle points?
I also referred to another post in this form with a very similar problem, except it was (e^x)(cosy) and it was determined to have no critical points.
Any ideas on this?
Thanks,
Kobbe
The Attempt at a Solution
I found the partials and mixed partial for the second derivative test as follows:
f[itex]_{x}[/itex]=-e[itex]^{x}[/itex](cos(y)-1)
f[itex]_{y}[/itex]=e[itex]^{x}[/itex](sin(y))
f[itex]_{xx}[/itex]=-e[itex]^{x}[/itex](cos(y)-1)
f[itex]_{yy}[/itex]=e[itex]^{x}[/itex](cos(y))
Knowing this, and that e[itex]^{x}[/itex] does not equal 0, then the critical points are periodic at 2∏n, where n is even intervals.
However, I get inconclusive when plugging it all into the second derivative test. And a quick query in WolframAlpha shows that there are indeed critical points, however no local maxima,minima or saddle points?
I also referred to another post in this form with a very similar problem, except it was (e^x)(cosy) and it was determined to have no critical points.
Any ideas on this?
Thanks,
Kobbe