I don't understand how to prove the following:
Use the Mean Value Theorem to show that
$|\cos(x) - \cos(y)| \leq |x-y| $
Why I'm confused:
(a)This question is from a single variable calculus book, but it seems like a multivariable problem. But I guess there must be implicit differentiation involved somehow.
(b) Mean Value Theorem involves an interval, and the "average rate of change" (ie the slope formula). However there is no interval, and I don't understand how slope can be used to show one multivariable function is less than another. (ex $2x+y+1 \gt 2x+y$, but they have the same slope)
(c) I have no idea how to deal with the absolute value here. Usually in a single variable calculus problem, I would break up an absolute value function into a 2 part piecewise function. Do I have to consider all 4 possible cases (++)(+-)(-+)(--)?
I have some ideas about how to prove it, but none of them involve the Mean Value Theorem.