1. 4.3
Using Derivatives for Curve Sketching

2. Absolute maximum
(also local maximum)
Local maximum
Local minimum
Local extremes are also called relative extremes.
Absolute minimum

3. Absolute maximum
(also local maximum)
Local maximum
y’ DNE
Local minimum
Notice that local extremes in the interior of the function occur where is zero or is undefined.

4. Critical Point:
A point in the domain of a function f at which
or does not exist is a critical point of f .
Note:
Maximum and minimum points in the interior of a function always occur at critical points, but critical points are not always maximum or minimum values.

5. Critical points are not always extremes!
(not an extreme)

6. (not an extreme) p

7. Point of Inflection
A point in the domain of a function f at which
or does not exist is a POSSIBLE point of inflection of f .
Note:
For a point to be an inflection point y’’ must change sign at that point.
How do we find points of inflection using y’?

8. X- values where f’’ = 0 or DNE are not necessarily points of inflection!
y = x4
y’’ = 0 but y does not change concavity at x = 0

9. X- values where f’’ = 0 or DNE are not necessarily points of inflection!
y = x2/3
y’’ DNE at x = 0 but y does not change concavity at x = 0

10. A couple of somewhat obvious definitions:
A function is increasing over an interval if the derivative is always positive.
A function is decreasing over an interval if the derivative is always negative.

11. First derivative:
is positive
Function is increasing.
is negative
Function is decreasing.
is zero or
undefined
Possible local maximum or minimum.
Second derivative:
is positive
Curve is concave up.
is negative
Curve is concave down.
is zero or
undefined
Possible inflection point
(where concavity changes).

12. This is the graph of y = f’(x), the derivative of f(x).
On what interval is f(x) increasing?
(b) On what interval is f(x) decreasing?

13. This is the graph of y = f’(x), the derivative of f(x).
(c) For what values of x does f have a relative maximum? Why?
(d) For what values of x does f have a relative minimum? Why?

14. This is the graph of y = f’(x), the derivative of f(x).
(e) On what intervals is f concave upward? Use f’ to explain.
(f) On what intervals is f concave downward? Use f’ to explain.

15. This is the graph of y = f’(x), the derivative of f(x).
(g) Find the x-coordinate of each point of inflection of the
graph of f on the open interval (-3,5). Use f’ to justify your answer.

17. Stop here G 

18. First derivative test:
Example:
Graph
There are roots at and
Possible extreme at
Set
First derivative test:
negative
positive
positive

19. First derivative test:
Example:
Graph
There are roots at and
Possible extreme at
Set
First derivative test:
maximum at
minimum at

20. Or you could use the second derivative test:
Example:
Graph
There are roots at and
Possible extreme at
Set
Or you could use the second derivative test:
negative
concave down
local maximum
positive
concave up
local minimum
maximum at
minimum at

21. Possible inflection point at .
Example:
Graph
We then look for inflection points by setting the second derivative equal to zero.
Possible inflection point at
negative
positive
inflection point at

22. p Make a summary table: rising, concave down local max
falling, inflection point
local min
rising, concave up p

23. Let f(x) = x3 – 12x – 5. Use the first derivative test to find the local extreme values.

24. Let f’(x) = 4x3 – 12x2.
Identify where the extrema of f occur.
Find the intervals on which f is increasing and decreasing.
Find where the graph of f is concave up and where it is
Concave down.
(d) Find any points of inflection.

25. Extreme Value Theorem:
If f is continuous over a closed interval, then f has a maximum and minimum value over that interval.
Maximum & minimum
at interior points
Maximum & minimum
at endpoints
Maximum at interior point, minimum at endpoint

26. Extreme values can be in the interior or the end points of a function.
No Absolute
Maximum
Absolute Minimum

27. Absolute
Maximum
Absolute Minimum

28. Absolute
Maximum
No Minimum

29. No
Maximum
No Minimum

30. Finding Maximums and Minimums Analytically:1
Find the derivative of the function, and determine where the derivative is zero or undefined. These are the critical points. 2
Find the value of the function at each critical point. 3
Find values or slopes for points between the critical points to determine if the critical points are maximums or minimums. 4
For closed intervals, check the end points as well.

31. Finding Maximums and Minimums Analytically:
Find the extrema of f(x) = 3x4 – 4x3 on the interval [-1,2]