4. Suppose that you know following properties of the function f(x): Use the prop

Question

4. Suppose that you know following properties of the function f(x): Use the properties above to analyze f(x) as outlined in (i)-(ix) above. a. Determine the equations of all vertical and horizontal asymptotes. If there are no asymptotes of a certain kind, say so explicitly. b. Determine the intervals where f(x) is increasing and the intervals where f(x) is decreasing. c. Determine the x-coordinates where f(x) has a local maximum and those where it has a local minimum. If there is no such minimum/maximum, say so explicitly. d. Determine the intervals where f(x) is concave up and the intervals where f(x) is concave down. e. Determine where the points of inflection occur. You may give the x-coordinate only. f. Sketch the graph of a function displaying the properties above. Take care to mark clearly all intercepts, asymptotes, local extrema, and points of inflection.

Explanation / Answer

(a):

vertical asymptotes :

x = 1

horizontal asymptotes:

y=3

y=-2

(b):

increasing:

x belongs to (-2,1) union (1, 3)

decreasing

x belongs to (-infinity , infinity) -  (-2,1) union (1, 3) union {1}

(c):

x = -2 , local minima

x = 3, local maxima

/(d):

concave up

x belongs to (-2,1) union (3, infinity)

concave down

x belongd to (-infinity , infinity) - (-2,1) union (3, infinity)

(e):

x = -4

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