Answer:
Below...
Step-by-step explanation:
To find the critical points, we need to find where the gradient of f(x,y) is equal to zero.
∇f(x,y) = (-sin x, sin y)
Setting each component of the gradient equal to zero, we get
-sin x = 0 and sin y = 0
This gives us critical points at (nπ, mπ) for any integers n and m.
Next, we need to find the second partial derivatives of f(x,y) at each critical point.
f_xx = -cos x
f_yy = cos y
f_xy = 0
At a critical point (nπ, mπ), we have f_xx = (-1)^n and f_yy = (-1)^m.
If f_xx and f_yy have the same sign, then the critical point is a local extremum. If they have opposite signs, then the critical point is a saddle point.
Thus, we have the following cases:
(nπ, mπ) is a local minimum if n and m are both even.
(nπ, mπ) is a local maximum if n and m are both odd.
(nπ, mπ) is a saddle point if n and m have different parities.
To find the level curves for z = 0, we set f(x,y) = 0:
cos y - cos x = 0
cos y = cos x
y = ±x + 2πk, where k is an integer
Thus, the level curves for z = 0 are given by the planes y = x + 2πk and y = -x + 2πk for integer values of k.