f(x, y) = 3ycos x, 0 ≤ x ≤ 2π

Finding the locla minima, maxima, and saddle points. Having difficulty as so far I have:
Fx = -3ysinx
Fy = 3cosx
Fxx = -3ycosx
Fyy = 0
Fxy = -3sinx

I have set both Fx and Fy equal to 0 but am not too sure about the critical points, which are my biggest problem. I fell like they would be set to x = 0, pi and y = 0 for Fx, and for Fy would be x = pi/2 and 3pi/2.

To find the local minima, maxima, and saddle points of the function f(x, y) = 3ycos(x), you need to identify the critical points and then analyze their nature using the second derivative test.

To find the critical points, set the partial derivatives Fx and Fy equal to zero and solve for x and y:

Fx = -3ysin(x) = 0
Fy = 3cos(x) = 0

From Fx = -3ysin(x) = 0, you have two cases: y = 0 or sin(x) = 0.

1. When y = 0, Fx becomes 0. So, for this case, the critical points occur when y = 0 and any value of x (assuming it satisfies the function's domain).

2. When sin(x) = 0, x can take the values 0, π, 2π, etc. To determine the corresponding y-values, substitute x into Fy = 3cos(x):

For x = 0, π, 2π, etc.: Fy = 3cos(0) = 3, Fy = 3cos(π) = -3, Fy = 3cos(2π) = 3, and so on.

So, the critical points are:
- When y = 0, x can be any value within the domain [0, 2π].
- When (x, y) = (0, 3), (π, -3), (2π, 3), etc.

Now, to determine the nature of these critical points (local minima, maxima, or saddle points), we need to use the second derivative test.

Calculate the second partial derivatives Fxx, Fyy, and Fxy:

Fxx = -3ycos(x)
Fyy = 0
Fxy = -3sin(x)

For each critical point, evaluate these second partial derivatives and apply the second derivative test:

1. For the critical points where y = 0:
- When x = 0, π, 2π, etc.:
- Fxx = 0
- Fyy = 0
- Fxy = 0

Since both Fxx and Fyy are zero, the second derivative test is inconclusive. Further analysis is needed.

2. For the critical points (0, 3), (π, -3), (2π, 3), etc.:
- Fxx = -3ycos(x)
- Fyy = 0
- Fxy = -3sin(x)

Plug in the values of x and y for each critical point and evaluate Fxx, Fyy, and Fxy. Then apply the second derivative test to determine the nature of each critical point.

After analyzing the second derivatives, you can determine whether each critical point is a local minimum, local maximum, or saddle point. If the second derivative test is inconclusive, you may need to explore other methods or techniques to further analyze those points.