016A Homework 6 SolutionJae-young Park∗October 6, 2008• 2.1 *4 Which functions have the property that the slope always de-creases as x increases?Solution (a), (e). You can find these answers by finding the graphswhich is convave down everywhere. For (b),(d),(c), the slope increasesas x increases. For (e), we have an inflection point.• 2.1 *6 Describe the graph.(See p.150)Solution It has a maximum at a ≈ −0.5 Increasing for x < a ≈−0.5, relative maximum at x = a ≈ −0.5 with maximum value α ≈5.2. decreasing for x > a. Concave down for x < 3 and inflection pointat x = 3, Concave up for 3 < x. x-intercept at (−3.5, 0), y-intercept at(0, 5.1). (y = 0 is an asymptote? Since we don’t have an informationfor 8 < x, this is not clear. You may say we have an asymptote ornot. Either way should be fine.) (Every value other than x = 3 is anapproximate)• 2.1 *10 Describe the graph. (see. p. 150)Solution Increasing for all x. no relative maximum or minimum.Concave down for x < 3, inflection point at x = 3. Concave up for3 < x. x-intercept at (−.5, 0) (approximate), y-intercept at (0, 1).Defined for all x and no asymptote.• 2.1 *12 Describe the graph. (see. p. 150)Solution Increasing for x < a ≈ −1.5, relative maximum at x =a ≈ −1.5 with maximum value ≈ 3.4, decreasing for a ≈ −1.5 < x <∗jaypark at m a t h . b e r k e l e y . e d u. GSI for 16A 101,104,1051b ≈ 2, relative minimum at x = b ≈ 2 with minimum value ≈ −1.5,increasing for b(≈ 2) < x < c ≈ 5.5, relative maximum at x = c ≈ 5.5with maximum value ≈ 3.3, decreasing for c(≈ 5.5) < x. Concavedown for x < 0, inflection point at (0, 1), concave up for 0 < x < 4,inflection point at (4, 1), concave down for 4 < x. x- intercepts are(−2.8, 0), (0.5, 0), (3.5, 0), (6.8, 0) y-intercept is (0, 1).• 2.1 *18 See p. 151 Figure 20.Solution (a) At which labeled points is the function decreasing?A,E (b) At which labeled points is the graph concave down?Assuming the points C,E are the inflection points, the answer is D.(c) Which labeled point has the most negative slope(that is, negativeand with the greatest magnitude)?At A,E , we have negative slopes. By inspecting, the point E has themost negative slope.• 2.1 *28 One method of determining the level of blood flow through thebrain requires the person to inhale air containing a fixedd concentra-tion of N2O, nitrous oxide. During the first minute, the concentrationof N2O in the jagular vein grows at an increasing rate to a level of.25%. Thereafter it gorws at a decreasing rate and reaches a concen-tration of about 4% after 10 minutes. Draw a possible graph of theconcentration of N2O in the vein as a function of time.Solution Look at other file.• 2.1 *30 Figure 22 gives the US electrical energy production in trillionkilowatt-hours from 1935 (t = 0) to 1995 (t = 60) with projections. Inwhat year was the level of production growing at the greatest rate?Solution The growing rate of the level of production is the slopeof the tangent line of the given graph. The slope is increasing for0 < t < 40, and attains its maximum at t = 40, which is the inflectionpoint and is decreasing after that point. Since t = 40 is equivalentto 1975, we conclude that the level of production was growing at thegreatest rate in 1975.• 2.1 *40 Suppose the function f(x) has a relative minimum at x = aand a relative maximum at x = b. Must f(a) be less than f (b)?Solution No, it doesn’t have to be that way. Look at other file.• Problem 1 Suppose that f(x) is differentiable and increases whenx < 8, decreases when 8 < x < 10, has an inflection point at x=92and increases when x > 10. Does f (x) have any relative maxima orminima? Where?Solution Relative maxima is where the graph changes from in-creasing to decreasing. So we have a relative maxima at x = 8. Sim-ilarly, relative minima is where the graph changes from decreasing toincreasing. So we have a relative minima at x = 10.• Problem 2 Let g(x) = f(x) + 83, where f (x) was defined in thelast question. What properties does g(x) have, i.e. where does itincrease, decrease, have any inflection points, relative minima, andrelative maxima? Justify your answers.Solution All the listed properties are invariant under parallel trans-lation along y-axis. So, g(x) increases when x < 8, decreases when8 < x < 10, has an inflection point at x = 9 and increases whenx > 10. We have a relative maxima at x = 8 and a relative minima atx = 10.• Problem 3 Let h(x) = −g(x), where g(x) was defined in the lastquestion. What properties does h(x) have, i.e. where does it increase,decrease, have any inflection points, relative minima, and relative max-ima? Justify your answers.Solution The graph of h(x) is the reflection about x axis of thegraph of h(x). So, h(x) decreases when x < 8, increases when 8 < x <10, has an inflection point at x=9 and decreases when x > 10. Wehave a relative minima at x = 8 and a relative maxima at x = 10.• Problem 4 Let f (x) = (x − 1)n, where n is a positive integer.For which positive integer values of n does f (x) have a criticalpoint? Where is it?Solution The critical point (a, f (a)) is where we have f0(a) = 0.Since f0(x) = n(x − 1)n−1, unless n = 1, we have f0(1) = 0. So wehave a critical point (1, 0) unless n = 1.For which positive integer values of n does f (x) have a relativeextremum? Where is it? Is is a relative maximum or minimum?Solution The critical point above are relative extremum whenf0(x) changes sign. If n is even, f0(x) changes sign and have a rel-ative minima. If n is odd, f0(x) is nonnegative. So we don’t have arelative extremum.3For which positive integer values of n does f (x) have an inflectionpoint? Where is it?Solution The inflection point is where the concavity of the graphchanges. We can check it by sketching graph. If n is even, the graph isconcave up. If n is odd and n 6= 1, the concavity of the graph changesat x = 1.• 2.2 *2 Which function have a negative first derivative for all x?Solution We have to find the graphs which is decreasing for all xand have no critical points. Considering the graphs, we get (b),(c),(f).• 2.2 *4 Which function have a negative second derivative for all x?Solution If the second derivative is negative for all x, the slope isdecreasing, so we have concave down. So (f).• 2.2 *6 Which one of the graph in Fig 17 could represent a functionf(x) for which f (a) = 0, f0(a) < 0, f00(a) > 0?Solution (a) and (c) have f (a) = 0. But for (a), f0(a)