A calculus problem by Jose Sacramento
True or False?
For all real and , the inequality is is fulfilled.
This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try
refreshing the page, (b) enabling javascript if it is disabled on your browser and,
finally, (c)
loading the
non-javascript version of this page
. We're sorry about the hassle.
1 solution
In the interval [a,b] for f(x)=sin(x) ; by Langrange Mean Value Theorem ; we have that there exists c (a<c<b) such that we have: f '(c)=(f(b)-f(a))/(b-a).Since we have our function as sin(x), we get the following: cos(c)=(sin(b) - sin(a))/(b-a).Using the fact that |cos(c)| is always less than or equal to 1, we get that |(sin(b) - sin(a))| is always less than or equal to |(b -a)| which is the result which we had to prove.