5.1 - Analysis of Functions
Okay, let’s analyze some functions! A few of these things we’ve gone over before, but we’re going
to go over them again, because they are very important in this chapter.
Let f be a continuous function on [a,b], and diﬀ on (a,b). Then if f is
Increasing on [a,b]: f (x) > 0 on (a,b).
Decreasing on [a,b]: f (x) < 0 on (a,b).
Constant on [a,b]: f(x) = 0 on (a,b).
1 3 2
Example: Find the intervals on which f(x) = x − 2x 3 3x − 4 is increasing and decreasing.
Solution: f (x) = x − 4x + 3 = (x − 3)(x − 1). The idea with this type of problem is to see when
f (x) = 0. The only times f (x) will change signs (because that’s what we’re lookinig for) are
when f (x) = 0 or undeﬁned. Well, this is a polynomial, which means there are no vertical
asymptotes or any other problems, so we should be ﬁne! So, f (x) = 0 when x = 1,3. Now all we
need to do is check the signs!
+++++++ 1 −−−−−−− 3 +++++++
0 2 4
All I did here was plug in 0, 2, and 4 into f (x) and looked to see if they were positive or negative.
Remember, f (x) positive means f is increasing. So, f is increasing on the intervals (−∞,1] and
on [3,∞). Meanwhile, f is decreasing on the interval [1,3].
Example: Find the intervals on which f(x) = x +x+7 is increasing and decreasing.
Solution: Well, let’s take that derivative again!
(x − 1)(2x + 1) − (x + x + 7) x − 2x − 8
f (x) = =
(x − 1)2 (x − 1)2
And, well, the top factors into (x + 2)(x − 4), so that’s when f (x) = 0, but we also can’t forget
about when the derivative is not deﬁned (VA’s), because sign changes may in fact occur there!
This would be when x = 1. So, let’s do that number line again!
+++++++ −2 −−−−−−− 1 −−−−−−− 4 +++++++
−3 0 2 5
So, f is increasing on the intervals (−∞,−2] ∪ [4,∞) and decreasing on (−2,1) ∪ (1,4). Notice
how we still have to throw out x = 1 from that interval of decreasing because it’s not in our
original domain of f(x)!
We shall now take a closer look at the second derivative. The concavity of a function is the study
of its curvature. There are two basic types of concavity: concave up and concave down. To
determine if a function f is concave up or down, we just take a look at the sign of the second
derivative at that point.
If f (x) > 0, then f is concave up (usually abbreviated CCU)
If f (x) < 0, then f is concave down (usually abbreviated CCD)
The only times that f can change concavity is when f (x) = 0 or f (x) is not deﬁned (ie, VA).
If f changes concavity at x an0 f (x ) = 0, then x is cal0ed an inﬂection point.
What does concave up look like? It looks like a giant smiley face. It’s happy to be concave up.
Or, if you want to think in terms of functions, it basically looks like y = x . Notice that a function
can be both increasing and decre