## Practice Problems

Work through the odd-numbered problems 1-17. Once you have completed the problem set, check your answers.

### Problems

In problem 1, each quotation is a statement about a quantitity of something changing over time. Let represent the quantity at time . For each quotation, tell what represents and whether the first and second derivatives of are positive or negative.

1. (a) "Unemployment rose again, but the rate of increase is smaller than last month".

(b) "Our profits declined again, but at a slower rate than last month".

(c) "The population is still rising and at a faster rate than last year".

3. Sketch the graphs of functions which are defined and concave up everywhere and which have

(a) no roots.

(b) exactly 1 root.

(c) exactly 2 roots.

(d) exactly 3 roots.

5. On which intervals is the function in Fig. 12

(a) concave up?

(b) concave down?

**Fig. 12**

In problems 7 and 9, a function and values of so that are given. Use the Second Derivative Test to determine whether each point is a local maximum, a local minimum or neither

11. At which labeled values of in Fig. 13 is the point an inflection point?

**Fig. 13**

13. How many inflection points can a

(a) quadratic polynomial have?

(b) cubic polynomial have?

(c) polynomial of degree n have?

15. Fill in the table with "", "", or "" for the function in Fig. 16

**Fig. 16**

17. Some people like to think of a concave up graph as one which will "hold water" and of a concave down graph as one which will "spill water". That description is accurate for a concave down graph, but it can fail for a concave up graph. Sketch the graph
of a function which is concave up on an interval, but which will not "hold water".

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-4.4-Second-Derivative.pdf

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