Since the focus of our attention here is on what makes the First Derivative Test true we will restate it to reflect our new, and deeper, understanding. Notice that the conclusion is the same, only the conditions have changed.
Suppose \(f(x)\) is continuous on the interval \([\alpha,\beta]\text{,}\) and differentiable on the interval \((\alpha,\beta)\text{.}\) Suppose further that both \(a\) and \(b\) are in the interval \([\alpha,\beta]\) and \(b\gt a\text{.}\)
If \(f^\prime(x)\gt 0\) on the interval \((\alpha,\beta)\) then \(f(b)\gt f(a)\text{.}\) (That is, the function is increasing on \([\alpha,\beta]\text{.}\))
If \(f^\prime(x)\lt 0\) on the interval \((\alpha,\beta)\) then \(f(b)\lt f(a)\text{.}\) (That is, the function is decreasing on \([\alpha,\beta]\text{.}\))
We want to use the Mean Value Theorem on the interval \([a,b]\) so we begin by verifying that the conditions of the Mean Value Theorem are satisfied on that interval. Observe that \([a,b]\) is a subinterval of \([\alpha,\beta]\) so \(f(x)\) is continuous on \([a,b]\) and differentiable on \((a,b)\text{.}\) By the Mean Value Theorem there is a number, \(c\text{,}\) in the interval \((a,b)\) such that
While we are in this frame of mind, we’ll take a moment to notice that we can use the Mean Value Theorem to prove, rigorously, something that we have alluded to a few times but have never addressed directly. It is clear from our differentiation rules that if two functions differ by a constant, then they have the same derivative. We’ve mentioned that the converse is true, namely if two functions have the same derivative on an interval then they must differ by a constant. This can be proved in a manner similar to the proof above.
Suppose \(f^\prime(x)=0\) on the interval \((\alpha,\beta)\) and that \(a\) and \(b\) are two points in that interval. Use an argument similar to the proof of the first derivative test to show that \(f(a)=f(b)\text{.}\)
We did not prove L’Hôpital’s Rule 12.4.25 in Chapter 12 because the general theorem is slightly beyond the scope of this text. But the somewhat simpler special case of Theorem 12.4.15, is a straightforward consequence of the Mean Value Theorem.
