Chapter 1, Problem 38
elishapeterson 09 Feb 2008 19:04
Hint 1: How do you write down the set of x values at which f evaluates to v? The answer is $X=\{x\in [a,b] : \quad?????\quad\}$ for some ??????.
Hint 2: What you are trying to show is that $X$ has both a minimum and maximum. Remember that these are x values!
Hint 3: How do you get at minimum/maximum? We can define the Least Upper Bound of $X$ and we know that exists, by the LUB property of the real numbers. Keep in mind, though, that while $LUB(X)$ always exists, it may not be in the set $X$.
Hint 4: So you can define an x-value $x'=LUB(X)$, and if you can show that $x'\in X$, then you've shown that $x'$ is the maximum of the set.