Section 2.8 What do f ' and f " say about f?
Definition: A function f(x) is increasing if whenever a<b , f(a) < f(b).
A function is decreasing if whenever a<b, f(b) < f(a).
If a function is differentiable on [a, b] then the following are true:
If f ' > 0 on [a, b] then f is increasing of [a, b]. If f ' < 0 on [a, b] then f is decreasing on [a, b].
If f " > 0 on [a, b] then f is concave up on [a, b]. If f " < 0 on [a, b] then f is concave down on [a, b].
How do we find the signs of f ' and f" ?
Example: Given that f ' (x) = x- 2 , where is f increasing ? decreasing? x-2 is positive only for x>2 so f is increasing on (2, ∞] and decreasing on (-∞, 2).
Example: Given that f'(x) = (x-2)(x-3), where is f increasing? decreasing?
We have to find the intervals where f ' is positive.
One way is to graph f ' and see where f ' is above the x-axis. (x-2)(x-3) is a parabola opening upward with zeros at x=2 and x=3 so it is positive on (-∞, 2) and (-∞, 3) and this is where f is increasing. f' is negative on (2, 3) and this is where f is decreasing.
Another way is to make a sign chart for f '.
The sign of f ' can only change if x-2 changes sign or if x-3 changes sign. x-a can only change sign at a.
Each of these does change sign once. If we test the sign at x=0, then change at x=2 and change it again at x=3 we get the following sign chart. f ' (0)=6 is positive so we start with +. signf ' +++++ 2 ------- 3 +++++
In the next example f ' has a factor that is squared, so that factor will not change the sign of f '.
Example: Given f ' ( x )

( x

2 )
2
( x

3 ) make a sign chart for f '. Again, x=0 is the most convenient point to test. f '(0)= -12 so we start out negative. The sign does not change at 2, because (x-2)is squared, but it does change at 3. sign f ' ---- 2 ----- 3 ++++++ So f is decreasing on (-∞, 3) and increasing on (3, ∞).
Example: f ' ( x )

( x

1 )
2
( x

2 )
3
THe only x-values where f ' can change sign are -1, 2, and 5.
( x

5 )

The factor ( x +1) does not change sign but ( x

2 )
3 and x -5 both do change sign. f '(0)= -8/(-5) is positive so we put a + between -1 and 2 where 0 is. Then change the signs as we cross over 2 and 5 but not as we cross over -1. sign f ' ++++++ -1+++++ 2 -------- 5 ++++++