# Proof using Lagrange M Mean Value Theorem

• Jan 18th 2010, 06:50 PM
amm345
Proof using Lagrange M Mean Value Theorem
Using Lagrange Mean Value Theorem prove that for every
a,b are real numbers on has |sina-sinb|<or=|a-b|
• Jan 18th 2010, 06:54 PM
Drexel28
Quote:

Originally Posted by amm345
Using Lagrange Mean Value Theorem prove that for every
a,b are real numbers on has |sina-sinb|<or=|a-b|

Let $a,b\in\mathbb{R}$. We know by the MVT that $\frac{\sin(a)-\sin(b)}{a-b}=f'(c)\implies \left|\sin(a)-\sin(b)\right|\leqslant |f'(c)||a-b|$ for some $c\in(a,b)$. But, $f'(c)=\cos(c)\leqslant1$ so that $\left|\sin(a)-\sin(b)\right|=\left|\cos(c)\right||a-b|\leqslant |a-b|$