## Practice Problems

Work through the odd-numbered problems 1-43. Once you have completed the problem set, check your answers.

### Problems

1. Label all of the local maximums and minimums of the function in Fig. 13. Also label all of the critical points.

**Fig. 13**

In problems 3-13, find all of the critical points and local maximums and minimums of each function.

15. Sketch the graph of a continuous function $\mathrm{f}$ so that

(a) , and the point is a relative maximum of .

(b) , , and the point is a relative minimum of .

(c) ,
is not differentiable at , and the point is a relative maximum of .

(d) , is not differentiable at , and the point is a relative minimum of .

(e) , and the point is not a relative minimum or maximum of .

(f) , is not differentiable at , and the point is not a relative minimum or maximum of .

In problems 17-25, find all critical points and local extremes of each function on the given intervals.

**Fig. 15**

29. Find the value for so the box in Fig. 17 has the largest possible volume? The smallest volume?

**Fig. 17**

31. Suppose you are working with a polynomial of degree on a closed interval.

(a) What is the largest number of critical points the function can have on the interval?

(b) What is the smallest number of critical points it can have?

(c) What
are the patterns for the most and fewest critical points a polynomial of degree on a closed interval can have?

33. Suppose and . What can we conclude about the point if

(a) for , and for ?

(b)
for , and for ?

(c) for , and for ?

(d) for , and for ?

35. is a continuous function, and Fig. 18 shows the graph of

(a) Which values of are critical points?

(b) At which values of is a local maximum?

(c) At which values of
is a local minimum?

**Fig. 18**

37. State the contrapositive form of the Extreme Value Theorem.

39. Imagine the graph of . Does have a **minimum** value for in the interval ?

(a)

(b)

(c)

(d)

(e)

41. Imagine the graph of . Does have a **minimum** value for in the following intervals?

(a)

(b)

(c)

(d)

(e)

43. Define to be the **slope** of the line through the points and , in Fig. 21 .

(a) At what value of is minimum?

(b) At what value
of is maximum?

**Fig. 21**

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-4.1-Finding-Maximums-and-Minimums.pdf

This work is licensed under a Creative Commons Attribution 3.0 License.