Matthews' Home MNWest Home Pre-Engineering D2L Site Index

MATH 1121.81 Calculus I

Chapter 3:

Applications of Derivatives

The First Derivative Test (something you already know!):

First, four official definitions:

Let f(x) be any function, and (a , b) an interval for which f(x) is defined.  Then,

1.  We say that f(x) is strictly increasing on (a , b), if, for all x1, x2 in (a , b), x1 < x2 -> f(x1) < f(x2).

2.  We say that f(x) is strictly decreasing on (a, b), if, for all x1, x2 in (a , b), x1 < x2 -> f(x1) > f(x2).

3.  We say that f(x) is monotonically increasing on (a , b), if, for all x1, x2 in (a , b), x1 < x2 -> f(x1) <= f(x2).

4.  We say that f(x) is monotonically decreasing on (a, b), if, for all x1, x2 in (a , b), x1 < x2 -> f(x1) >= f(x2).

(Strictly Increasing is also sometimes called "order-preserving," while Strictly Decreasing is also sometimes called "order-reversing.")

So, in common English, we're talking about "going up" from left to right, or "going down" from left to right, if you're looking at the graphs of functions.

Now, here's the first common sense bit:  "Going UP" means the first derivative is positive, and "Going DOWN" means the first derivative is negative (just like slopes of lines!):

Here's the theorem, technically stated:

To summarize the various cases:

Given f(x) differentiable on an interval (a , b):

If, for all x in (a , b):  

Then

f(x) Is Strictly Increasing  

f ' (x) >= 0

f(x) Is Strictly Decreasing  

f ' (x) <= 0

f(x) Is Constant  

f ' (x) = 0

f ' (x) > 0  

f(x) Is Strictly Increasing

f ' (x) < 0  

f(x) Is Strictly Decreasing

f ' (x) = 0  

f(x) Is Constant

Putting the above result together with a bit of common sense, we end up with the famous "First Derivative Test" for functions (which implies that there will at least be a "Second Derivative Test," and there will):

But, again, we need another definition, that of "local extreme point:"

Let f(x) be any function defined on an interval, (a , b):

1.  f(x) is said to have a local (or relative) minimum at x = c in (a , b) if there exists some h > 0, such that:

for all x in (c - h, c) U (c, c + h), f(x) > f(c).

2.  f(x) is said to have a local (or relative) maximum at x = c in (a , b) if there exists some h > 0, such that:

for all x in (c - h, c) U (c, c + h), f(x) < f(c).

 

 The First Derivative Test:  Let f(x) be continuous and differentiable at a critical number, c. 

1.  If there exists a neighborhood, (c - h, c + h) such that for all x in (c - h, c), f ' (x) < 0, and for all x in (c, c + h) f ' (x) > 0, then f(x) has a local minimum at x = c.

2.  If there exists a neighborhood, (c - h, c + h) such that for all x in (c - h, c), f ' (x) > 0, and for all x in (c, c + h) f ' (x) < 0, then f(x) has a local maximum at x = c.

3.  If there exists a neighborhood, (c - h, c + h) such that for all x in (c - h, c), f ' (x) > 0, and for all x in (c, c + h) f ' (x) > 0, or such that for all x in (c - h, c), f ' (x) < 0, and for all x in (c, c + h) f ' (x) < 0,then f(x) has neither a local maximum nor a local minimum at x = c.

The applet makes the test seem "obvious," and, for now, I'll omit the proof.  The basic idea is "down, then up" equals minimum.  "Up, then down" equals maximum.

Applet:

The fact is, this theorem has less practical use than it did before the advent of graphing utilities, but there are still times where it can be useful in its own right; and there are other important results that build on it.

Next Time, Section 3.4 (Higher Derivatives):