Some Problems Involving I.V.T. November 25, 2008
Posted by putnam120 in Math Related, Uncategorized.trackback
Well I am in a big Analysis mood. I guess that it has something to do with the fact that at this point in my life Analysis is the topic in math that spikes my interest the most. This can be accredited to my Calculus BC teacher from high school (Mrs. Johnson) and my undergraduate Advanced Calculus professor (Dr. Shen). They both did a wonderful job of presenting material and showing applications and fascinating consequences.
Anyway onto the post, I was reading through the intermediate real analysis section of Problem Solving Through Problems. Here are some of the problems I found interesting.
(a) Suppose that ![f:\left[a,b\right]\to\mathbb{R}](http://s0.wp.com/latex.php?latex=f%3A%5Cleft%5Ba%2Cb%5Cright%5D%5Cto%5Cmathbb%7BR%7D&bg=ffffff&fg=666666&s=0 "f:\left[a,b\right]\to\mathbb{R}")
![g:\left[a,b\right]\to\mathbb{R}](http://s0.wp.com/latex.php?latex=g%3A%5Cleft%5Ba%2Cb%5Cright%5D%5Cto%5Cmathbb%7BR%7D&bg=ffffff&fg=666666&s=0 "g:\left[a,b\right]\to\mathbb{R}")
![x\in\left[a,b\right]](http://s0.wp.com/latex.php?latex=x%5Cin%5Cleft%5Ba%2Cb%5Cright%5D&bg=ffffff&fg=666666&s=0 "x\in\left[a,b\right]")
![\left[a,b\right]](http://s0.wp.com/latex.php?latex=%5Cleft%5Ba%2Cb%5Cright%5D&bg=ffffff&fg=666666&s=0 "\left[a,b\right]")
(b) Let ![f:\left[0,1\right]\to\mathbb{R}](http://s0.wp.com/latex.php?latex=f%3A%5Cleft%5B0%2C1%5Cright%5D%5Cto%5Cmathbb%7BR%7D&bg=ffffff&fg=666666&s=0 "f:\left[0,1\right]\to\mathbb{R}")
![\left[0,1-\frac{1}{n}\right]](http://s0.wp.com/latex.php?latex=%5Cleft%5B0%2C1-%5Cfrac%7B1%7D%7Bn%7D%5Cright%5D&bg=ffffff&fg=666666&s=0 "\left[0,1-\frac{1}{n}\right]")
(c) Assume the same conditions on 
![c\in\left[a,b\right]](http://s0.wp.com/latex.php?latex=c%5Cin%5Cleft%5Ba%2Cb%5Cright%5D&bg=ffffff&fg=666666&s=0 "c\in\left[a,b\right]")
My thoughts: The solution to (a) is a pretty straight forward application of the intermediate value theorem. For problem (b) I am considering looking at the slope of the line connecting 
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