Page 1 | © 2012 by Janice L. Epstein 5.1 □ f and f’ What does f ’ say about f ? (Section 5.1)
3
f
Page 2 | © 2012 by Janice L. Epstein 5.1 □ f and f’ EXAMPLE 1
The graph of the derivative of f is shown.
4
2
f’
3
2
1
1
-5
-4
-3
-2
-1
1
2
3
4
5
-1
-2
-1
1
2
-2
-1
-3
-4
-2
-3
Where is the function increasing? decreasing?
Where does the function have a local maximum? local minimum?
Where is the function concave up? concave down?
If
If
If
If
f ¢( x) > 0 on an interval, then f is increasing on the interval.
f ¢( x) < 0 on an interval, then f is decreasing on the interval.
f ¢¢( x) > 0 on an interval, then f is concave up on the interval.
f ¢¢( x) < 0 on an interval, then f is concave down the interval.
A critical number of a function f is a number c in the domain of f
such that either f ¢(c) = 0 or f ¢(c) does not exist.
An inflection point of a function f is the point where a function
changes concavity.
(a) Where is the function increasing or decreasing?
(b) Where might the function have a local maximum or minimum?
(c) Where is the function concave up or concave down?
(d) Where are the inflection points?
(e) If f (0) = 0 , sketch a possible graph of f.
Page 3 | © 2012 by Janice L. Epstein 5.1 □ f and f’ EXAMPLE 2
The graph of the derivative of f is shown.
f’
Page 4 | © 2012 by Janice L. Epstein -1
1
2
3
4
5
6
(a) Where is the function increasing or decreasing?
(b) Where might the function have a local maximum or minimum?
(c) Where is the function concave up or concave down?
(d) Where are the inflection points?
(e) If f (0) = 0 , sketch a possible graph of f.
5.1 □ f and f’ EXAMPLE 3
Sketch a graph of f satisfying the following conditions:
i.
f ¢( x ) > 0 on (-¥,1) and f ¢( x ) < 0 on (1,¥)
ii.
f ¢¢( x) > 0 on (-¥, -2) and (2,¥)
iii.
f ¢¢( x) < 0 on (-2, 2)
lim f ( x) = -2 and lim f ( x) = 0
iv.
x-¥
-2
x¥