January 4th, 2015, 08:59 PM  #1 
Senior Member Joined: Sep 2013 From: Earth Posts: 827 Thanks: 36  Intermediate Value Theorem
Use the Immediate Value Theorem to show that there is a root of the given equation in the specified interval. tan(x)=2x (0,1.4) How to solve this question? Last edited by greg1313; January 4th, 2015 at 10:02 PM. 
January 4th, 2015, 09:58 PM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,805 Thanks: 1045 Math Focus: Elementary mathematics and beyond 
It's intermediate value theorem, not "Immediate Value Theorem". Since one can find values where tan(x)  2x is negative and where tan(x)  2x is positive in the given interval, at some point in the given interval tan(x)  2x = 0 (by the intermediate value theorem). I don't know how much detail is required to provide an acceptable answer your question. 
January 4th, 2015, 10:16 PM  #3 
Newbie Joined: Jul 2012 From: Houston, Texas Posts: 18 Thanks: 2 Math Focus: Geometry , Modern Algebra 
I will give a try. Take the function $f(x) = \tan(x)  2$ We know that $f(x)$ is continuous on the interval $(0,1.4)$ Because this is an open interval, we can't choose 0 and 1.4. Pick two value x= $0.1$ and $x =1.39$ to test the sign of $f(x)$. At $x = 0.1, f(0.1) < 0$ At $x = 1.39, f(1.39) > 0$ The Intermediate Value Theorem states that if $f(a).f(b) <0$ then there exist an $x$ between$ (a,b)$ such that $f(x) = 0.$ The problem can also be solve by using Intermediate Value Theorem for Derivative. Last edited by skipjack; January 5th, 2015 at 06:10 AM. 
January 5th, 2015, 05:11 AM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,305 Thanks: 2443 Math Focus: Mainly analysis and algebra 
Strictly speaking, this is Bolzano's theorem which is a special case of the intermediate value theorem.


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