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.