Math 1441 Fall 2006 Sections 1 and 2 P. Achar 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 Collections: Mathematics