Is the function $f(x, y)= x\sin y+y\cos x $ Lipschitz on $[-a, a]\times[-b,b]?$

Question

My attempt

Let $(x, y_1)$ and $(x, y_2)$ be two points in the rectangle $R=[-a, a]\times[-b,b].$ Then

$$|f(x, y_1)- f(x, y_2)|\le|x||\sin y_1 -\sin y_2| + |y_1-y_2|$$

I can't understand how to proceed further; please give me some hints

Answer 1

Your function is differentiable. Thus, checking if your function is Lipschitz is just looking at your derivative and checking that it is bounded. Since your derivative is continuous and the set $[-a,a]\times[-b,b]$ is compact, the derivative is bounded. Thus, your function is Lipschitz.