1. Applications of Derivatives
Unit 4
Applications of Derivatives

2. Types of Maximums and Minimums
4.1 Critical Number Test
Types of Maximums and Minimums
Absolute/Global Extreme Values
Maximum –
Minimum –
Local/Relative Extreme Values

3. 4.1 Critical Number Test
Critical Number/Value –

4. 4.1 Critical Number Test
For the following picture, find the absolute extrema, local
extrema and critical numbers.

5. 4.1 Critical Number Test
Extreme Value Theorem – If f is continuous on a closed interval
[a, b], then f has on that interval.

6. Critical Numbers Test for Finding Extrema
4.1 Critical Number Test
Critical Numbers Test for Finding Extrema
If f is continuous on a closed interval [a, b], then it’s absolute extreme values are paired with critical numbers or endpoints.
Steps for finding the absolute extrema. 1. 2. 3. 4.

7. 4.1 Critical Number Test
Find the critical numbers of f. Then use the critical number test to determine the absolute extreme values of f. 1)

8. 4.1 Critical Number Test
Find the critical numbers of f. Then use the critical number test to determine the absolute extreme values of f. 2)

9. 4.2 Mean Value Theorem

10. 4.2 Mean Value Theorem
Determine if f satisfies the conditions of the mean value theorem. If so, find all possible values of c. 1)

11. 4.3 First Derivative Test

12. 4.3 First Derivative Test

13. 4.3 First Derivative Test

14. 4.3 First Derivative Test

15. 4.3 First Derivative Test

16. 4.4 Second Derivative Test
If the second derivative is positive, a curve looks like
If the second derivative is negative, a curve looks like

17. 4.4 Second Derivative Test
f is concave up on (a, b) if and only if the slopes of the tangents to
f are increasing
which means
f is concave down on (a, b) if and only if the slopes of the tangents
to f are decreasing

18. 4.4 Second Derivative Test
(c, f(c)) is an inflection point if and only if 1. 2.

19. 4.4 Second Derivative Test

20. 4.4 Second Derivative Test

21. 4.4 Second Derivative Test
Relationship of f’ and f’’ to f f

22. 4.4 Second Derivative Test