Question

Hi all, I am still working on second derivative tests having trouble with finding the critical points.
The problem is find the critical points of f(x,y)=sinx+siny+cos(x+y) for x between 0 and pie/4 and y between 0 and pie/4. Classify each as a local min, max, or saddle point.
f(x,y)=sinx+siny+cos(x+y)
I found the partials to be:
derivative of f with respect to x = fx=cosx-sin(x+y)
derivative of f with respect to y = fy=cosy-sin(x+y)

I am having trouble solving the partials for 0.

Answer 1

For
fx=cosx-sin(x+y)=0
fy=cosy-sin(x+y)=0

x=pie/6
y=pie/6

I did this by trial and error. Does anyone know how to solve it algebraically?

Answer 2

First, eliminate sin(x+y):
fx-fy=0
cos(x)-cos(y)=0
cos(x)=cos(y)
x=y

Plug x=y to either fx or fy:
fx=cos(y)-sin(y+y)=0
cos(y)=sin(2y)
cos(y)=cos(pi/2-2y)
y=(pi/2)-2y
3y=pi/2
y=pi/6

Since x=y,
x=pi/6

Answer 3

Very clever!! That makes a lot of sense!! It's amazing how easy it looks once you've seen it done. I have been puzzling over that one for days! Thanks!

Answer 4

Ur welcome! :)