You find yourself in a strange undulating landscape given by the function z = f (x, y) = cos y − cos x,
where z is the elevation.
1. Find all maxima, minima, and saddle points. What are the level curves for z = 0?
Graph this function.

Answers

Answer 1

Answer:

Below...

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.


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