4.2 Mean Value Theorem | Video 4 - Show Equation Has Exactly 1 Root || Calculus 1

in this video we will do an example in which we show an equation has exactly one root so the problem says show the equation tangent inverse of x minus two x plus one equals zero has exactly one root so there's exactly one x value that's going to make this function on the left-hand side to be

equal to zero all right so let's get into the proof so for the proof there's gonna be two parts to our argument we are gonna first show that it has at least at least one root and then later later we are going to show that it has at most one root at most one root and if we can do both

of those things so it has at least one root and at most one root combined those would imply that it has exactly one so that's going to be the structure of our argument okay so for showing it has at least one root we've done problems like this before in an earlier section so i want to give you an opportunity to

practice this pause the video for four minutes see if you can show this has at least one room four three two one pause it pause it and try for four minutes i'm going to give you a hint though the key idea to showing it has at least one root is to use ivt the intermediate value theorem

all right pause it for four minutes all right let's talk about it together now so i'm going to let this whole thing on the left hand side be my function i'm going to let f of x be tangent inverse of x minus 2x plus 1. and i want to show that this function equals 0.

so to do that i will need to be able i need to find two numbers i can plug in one that will make this whole function be less than zero one that will make the function be bigger than zero because the intermediate value theorem will then tell me oh it's got to be somewhere in