Finding Critical Points Of A Function

[Lancelot59]

I'm trying to find the minimia and maxima of the following function without using LaGrange multipliers:

$$f(x,y)=sin(x)+sin(y)+sin(x+y)$$
where:
$$0\leq x \leq 2\Pi$$
$$0\leq y \leq 2\Pi$$

Partial derivatives:
$$f_{x}=cos(y)+cos(x+y)$$
$$f_{y}=cos(x)+cos(x+y)$$

$$f_{xx}=-sin(x+y)$$
$$f_{xy}=-sin(y)-sin(x+y)$$
$$f_{yy}=-sin(x+y)$$

Now I have no clue how to get all the critical points. I simplified it using fx=0, fy=0, to get fx=fy. Equating and simplifying I got cos(x)=cos(y), x=y.

Now this is where I get lost. How do you pick points to try and solve the equations? I could say that cos(x)=cos(y), then cos(x)-cos(y)=0. So you could pick x or y to be pi/2 or 3pi/2.

However the solution manual to my textbook does not use any of these points...so how does this work?

 

[Mark44]

I'm trying to find the minimia and maxima of the following function without using LaGrange multipliers:

$$f(x,y)=sin(x)+sin(y)+sin(x+y)$$
where:
$$0\leq x \leq 2\Pi$$
$$0\leq y \leq 2\Pi$$

Partial derivatives:
$$f_{x}=cos(y)+cos(x+y)$$
$$f_{y}=cos(x)+cos(x+y)$$

You got off to a bad start. The two first partials are incorrect.
f_x = cos(x) + cos(x + y)
f_y = cos(y) + cos(x + y)

Because of this error, there are errors in your second partials as well.

$$f_{xx}=-sin(x+y)$$
$$f_{xy}=-sin(y)-sin(x+y)$$
$$f_{yy}=-sin(x+y)$$

Now I have no clue how to get all the critical points. I simplified it using fx=0, fy=0, to get fx=fy. Equating and simplifying I got cos(x)=cos(y), x=y.

Now this is where I get lost. How do you pick points to try and solve the equations? I could say that cos(x)=cos(y), then cos(x)-cos(y)=0. So you could pick x or y to be pi/2 or 3pi/2.

However the solution manual to my textbook does not use any of these points...so how does this work?

 

[Lancelot59]

You got off to a bad start. The two first partials are incorrect.
f_x = cos(x) + cos(x + y)
f_y = cos(y) + cos(x + y)

Because of this error, there are errors in your second partials as well.

WolframAlpha agrees with me.
Partial X
Partial Y

 

[Mute]

WolframAlpha agrees with me.
Partial X
Partial Y

Take a careful look at what you wrote for the first derivatives in your first post, what Mark44 wrote, and what wolframalpha is giving you. I assure you wolframalpha agrees with Mark44 (and that the error affects your second derivatives).

 

[Lancelot59]

DAH! That would do it...

So if fx=fy

cos(x)+cos(x+y)=cos(y)+cos(x+y)
cos(x)=cos(y)
x=y

Now what?

 

[Mute]

DAH! That would do it...

So if fx=fy

cos(x)+cos(x+y)=cos(y)+cos(x+y)
cos(x)=cos(y)
x=y

Now what?

Plug y=x back into f_x=0 to figure out what x (and hence y) must be. To determine if the point is a min or a max you need to recompute the partial derivatives, as the second derivatives in the first post are incorrect because you initially mixed up f_x and f_y. Then, calculate the value of the Hessian matrix to see if it's a min, max or saddle.

 

[Lancelot59]

I got it. Thanks!