 
Summary: Math 1441 Fall 2006
Sections 1 and 2 P. Achar
Exam 2 Solutions
October 12, 2006
Total points: 100 Time limit: 80 minutes
No calculators, books, notes, or other aids are permitted. You must show your work and justify your
steps to receive full credit.
1. Short answer:
(a) (4 points) What is the definition of critical point?
Solution: A point x where f (x) is either 0 or undefined.
(b) (4 points) Suppose f(x) is concave up on the interval (2, ), and that x = 3 is a critical point.
Which of the choices below to describes this critical point? Briefly explain your answer.
relative minimum relative maximum neither can't tell
Solution: Relative minimum. The fact that f(x) is concave up on (2, ) means that f (x) is
positive on (2, ). In particular, f (3) > 0, so the Second Derivative Test says that f(x) has a
relative minimum at x = 3.
(c) (4 points each) Sketch an example of a graph with the property that:
(i) it is increasing and concave down (ii) x = 0 is an inflection point but not a critical point
Solution: Solution:
4 2 2 4
