1. 5.2 Section 5.1 – Increasing and Decreasing Functions
The First Derivative Test and its documentation
5.2

2. The Theory First……
THE FIRST DERIVATIVE TEST
If c is a critical number and f ‘ changes signs at x = c, then
f has a local minimum at x = c if f ‘ is negative to the left
of c and positive to the right of c.
f has a local maximum at x = c if f ‘ is positive to the left
of c and negative to the right of c.

3. _ _ 1 3 -3 5 +
There is a rel min at x = 1 because f ‘ changes from neg to pos at 1
There is a rel max at x = 3 because f ‘ changes from pos to neg at 3

4. The Theory…Part II
EXTREME VALUE THEOREM
If a function f is continuous on a closed interval [a, b] then f
has a global (absolute) maximum and a global (absolute)
minimum value on [a, b].
GLOBAL (ABSOLUTE) EXTREMA
A function f has:
A global maximum value f(c) at c if f(x) < f(c) for every x in
the domain of f.
A global minimum value f(c) at c if f(x) > f(c) for every x in

5. The Realities…..
On [1, 8], the graph of any continuous function HAS to
Have an abs max
Have an abs min

6. _ +
There is an abs min at x = -1/2 because
f ‘ (x) < 0 FOR ALL x < -1/2 and
f ‘ (x) > 0 FOR ALL x > -1/2

7. Justify your answer. _ _ -2 -1 3 + +

8. _ +

9. Justify your answer. _ _ +

10. Justify your answer. _ +

11. GRAPHING CALCULATOR REQUIRED

12. 1.684
0.964

13. [0, 0.398), (1.351, 3]

14. The absolute max is 1.366 and occurs when x = 3
The absolute min is –0.098 and occurs when x = 1.351

15. Let k = 2 and proceed

16. _ + + 3 6

17. _ _ 1 +

18. CALCULATOR REQUIRED