Is the Mean Value Theorem Applicable to Prove sin x < x for x > 0?

[dontdisturbmycircles]

Homework Statement

Show that sin x < x for all x > 0

The Attempt at a Solution

I thought I was pretty good at calculus so I have kinda been shifting my calc class onto the bottom of my todo list, but this mean value theorem problem is giving me some problems.

For x > 1, sin x \leq 1 < x

This was my start... after about 25 minutes of thinking about how the mean value theorem could be applied I looked in the back.. (with much reluctance, trust me). They chose 2pi instead of 1 for the first part, that is likely significant but I am not getting thepoint. They then proceed to prove that (sinx)/(x) = cos(c) for some c in (0,2pi) and I can see that at that point c sin x< x since cos(c)\leq 1 but ONLY at that point c... I don't see how it proves it for the general case in that interval...

 

[NateTG]

Can you show that the slope of the tangent of \sin(x) is less than one for 0&lt;x \leq 1?

Then, if you assume that there is a point with x&gt;0,x=\sin(x) what happens when you apply the mean value theorem?