Dr. Byrne Math 125 Name____________________________________ 4.3/4.4 Comparing The First and Second Derivative Tests Home Work: Turn this in for a Quiz Grade Definition of a critical point: A number c in the domain of f (x) is called a critical point if f '(c) = 0 or f '(c) does not exist. Comments: The critical points of f (x) are always defined by the zeros (or undefined points) of the first derivative. In order to be a critical point, a point must be in the domain of f (x). Often points for which f '(x) is not defined are also points where f (x) is not defined, so we would exclude these. PROS AND CONS OF THE TWO TESTS: The first and second derivative tests can be used to identify a critical point as a minimum or a maximum. The first derivative requires testing the sign of TWO points around the critical point, but always provides a classification. The second derivative test requires testing the sign of ONE point around the critical point, but is sometimes inconclusive. 1st Derivative Test Let f (x) be differentiable in a neighborhood containing c. If c is a critical point, then if 푓′(푥) changes from + to − at c, 푓(푐) is a local maximum. if 푓′(푥) changes from − to + at c, 푓(푐) is a local minimum. if 푓′(푥) does not change sign at c, 푓(푐) is neither a minimum nor a maximum. 2nd Derivative Test Let f (x) be twice-differentiable in a neighborhood containing c. If c is a critical point, then if 푓′′(푐) > 0, 푓(푐) is a local minimum if 푓′′(푐) < 0, 푓(푐)is a local maximum if 푓′′(푐) = 0 or does not exist, the test is inconclusive (푓(푐) may be local max, local min or neither) (Note: An inflection point occurs where 푓′′(푥) exists and the concavity changes.) Example A: 푦 = 3푥 − 푥3 1) On the graph provided, label all maxima, minima and inflection points (max/min/I). 2) Find all critical points of y: 3) First derivative test: a. Identify the intervals where y is increasing or decreasing: y increasing: . y decreasing: . b. What does the first derivative test tell us about each critical point? (Review how the test works on the first page of this worksheet.) 4) Second derivative test: a. Identify the intervals where y is concave up or concave down: y concave up: . y concave down: . b. What does the second derivative test tell us about each critical point? Example B: 푦 = 푥5 − 5푥4 1) On the graph provided, label all maxima, minima and inflection points (max/min/I). 2) Find all critical points of y: 3) First derivative test: a. Identify the intervals where y is increasing or decreasing: y increasing: . y decreasing: . b. What does the first derivative test tell us about each critical point? 4) Second derivative test: a. Identify the intervals where y is concave up or concave down: y concave up: . y concave down: . b. What does the second derivative test tell us about each critical point? Example C: 푦 = 5푥2/5 − 2푥 1) On the graph provided, label all maxima, minima and inflection points (max/min/I). 2) Find all critical points of y: 3) First derivative test: a. Identify the intervals where y is increasing or decreasing: y increasing: . y decreasing: . b. What does the first derivative test tell us about each critical point? 4) Second derivative test: a. Identify the intervals where y is concave up or concave down: y concave up: . y concave down: b. What does the second derivative test tell us about each critical point? Example D: Comparing the 1st and 2nd derivative tests for classifying the critical point of 푓(푥) = 푥3 + 3. (a) Find the critical point of f (x). (b) What does the first derivative test tell us about this critical point? (c) What does the second derivative test tell us about this critical point? Example E: st nd ( ) 1 4 Comparing the 1 and 2 derivative tests for classifying the critical point of 푓 푥 = 4푥 + 1. (a) Find the critical point of f (x). (b) What does the first derivative test tell us about this critical point? (c) What does the second derivative test tell us about this critical point? Example F: Comparing the 1st and 2nd derivative tests for classifying the critical point of 푓(푥) = 푥(푥 − 1). (a) Find any critical points of f (x). (b) What does the first derivative test tell us about each critical point? (c) What does the second derivative test tell us about each critical point? Example G: 4 Comparing the 1st and 2nd derivative tests for classifying the critical point of 푓(푥) = 1 − . 푥2 (a) Find any critical points of f (x). (b) What does the first derivative test tell us about each critical point? (c) What does the second derivative test tell us about each critical point? .