Math. Problem 1 and 4 only.

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University of Oslo / Department of Economics four problem pages plus this, English only∗ ECON3120/4120 Mathematics 2 hand-in 2020 no. 4 DRAFT This draft will need proof-reading tonight, sorry for the delay. You can start working. Deadline to be announced shortly, will at the earliest be Thursday 5th. Math 2 hand-ins are pass/fail only. The typical pass mark for this course is forty percent, weighting at grader’s discretion, defaulting to uniform over letter-enumerated items. (Any grade indicated is for information only, and does not count towards your final grade.) Hand-in no. 4 will consist of Problem 4 (weight 2/3) and one of Problems 1, 2 or 3 (weight 1/3) that – if applicable – corresponds to the number of a hand-in you did not get approved: • If you have two out of three hand-ins approved, you shall submit Problem 4 (weight 2/3) and one of Problems 1, 2, 3, the number corresponding to the one you failed. For example, if you have hand-ins 1 and 3 approved but not number 2, you shall submit Problem 2 (weight 1/3) and Problem 4 (weight 2/3). • If you have one out of three hand-ins approved, then you might qualify for that “second- attempt”. To get hand-in no. 4 approved, you will submit as the previous bullet item; if you select to submit more than one of problems 1, 2, 3, we will select to your benefit. – for the “second-attempt”: please await further announcements to be made If you have hand-ins 1 through 3 approved, but would like to submit for feedback: please indicate on the front page that you have 3/3 already, that you would like to get feedback on [problem 4 and you can then choose]. • You are required to state reasons for all your answers. NEW INFORMATION COMPARED TO HAND-INS 1–3: – Answers must be justified as if you had no books, calculators or any other tools available; – You are allowed to use any tools that would be permitted for the exam, and refer to them – but no known theory needs a book reference; – Direct quotations from any source must be cited properly! (One obvious exception: you need not quote the problem text or anything else from this document, and if you do, you need not attribute it in any way.) • You are permitted to use any information stated in an earlier letter-enumerated item (e.g. “(a)”) to solve a later one (e.g. “(c)”), regardless of whether you managed to answer the former. Note that a later item need not require answers from or information given in a previous one. Problems 1, 2, 3, 4 are given on the respective pages 1, 2, 3, 4 to follow. ∗Answers can be given in English, Norwegian, Swedish or Danish, as per the regulations for the exam. 0 Problem 1 Let h(x, y, z) = e(x+y)z − xe1−y. a) (i) (x, y) = (−1, 1) is a stationary point for the function w(x, y) = h(x, y, 1). Classify it. (ii) It is a fact that h has a stationary point at (x, y, z) = (−1, 1, 1), and part (i) rules out some possibilities about the classification of this point. What possibility/possibilities is/are ruled out, and which remain(s)? For the rest of Problem 1, consider the maximization problems max h(x, y, z) subject to x+ y + z = 1 and z = Rx(L) max h(x, y, z) subject to x+ y + z ≤ 1, y ≥ 0 and 0 ≤ z ≤ Rx.(K) b) • State the Lagrange conditions associated with problem (L) • State the Kuhn–Tucker conditions associated with problem (K). • Pick one of the problems (K), (L) (your choice, one may be easier than the other), and check whether the extreme value theorem grants the existence of solution. c) In this part, let R = 1. Check whether the point (x̂, ŷ, ẑ) = (0, 1, 0) satisfies • the Lagrange conditions associated with (L) • the Kuhn–Tucker conditions associated with (K). (Hint: «None» is wrong answer.) d) If you verified precisly one one condition set in the previous part, select the corresponding problem; otherwise, choose freely between problems (K) and (L). If (x̂, ŷ, ẑ) is optimal for that problem when R = 1, approximately how much will the optimal value change when R is changed to 0.98? 1 Problem 2 Let q > 0 be a constant and define g(x) = e −x−2 |x|q . a) For each of the limits lim x→−∞ g(x), lim x→+∞ g(x), lim x→0− g(x), lim x→0+ g(x): Will l’Hôpital’s rule help you get a simpler limit to solve by hand? b) In this part you shall find those limits by considering ln g(x): • Explain why lim x→−∞ ln g(x) = lim x→+∞ ln g(x), and that they are < 0. • Show that lim x→0 g(x) = 0. Let now f(x) = g(x) + 2x 1 + x2 − 1 for x > 0. You can assume that f has at most two zeroes.† c) • Show that f has at least one zero z ∈ (0, 1) and at least one zero b > 1. • Show that f has a maximum x∗ and that x∗ ∈ (z, b). d) Both x∗, b and V = ∫ b x∗ f(x)dx depend on q. Find expressions for approximately how much they change if q decreases from e to 2.75. †If there are more, everything to follow will go through by letting z be the smallest and b the largest. But you can without any further considerations take for granted that there are no more. 2 Problem 3 Let for s > 0 f be the function f(s) = e −s−2 sq + 2s 1 + s2 − 1 where q > 0 is a constant. (Cf. problem 2.) a) (Carries half as much weight as part b).) • Show by by antidifferentiation that F (s) = 1 + s 2 2s2 e−s −2 − s + ln(1 + s2) is an anti- derivative for f when q = 5 (hint: substitution!), and • Calculate an antiderivative A when q = 3, and • pick some positive integer q > 5 (your choice!) and explain how to calculate an antiderivative for that case. b) (Carries twice as much weight as part a).) Let in this part q = 5 so that f(t) = e−t −2 t5 + 2t 1 + t2 − 1. For each of the four differential equations (D1)–(D4) below, find the particular solution which satisfies limt→0+ x(t) = z, where z ∈ (0, 1) is the zero for f as given in Problem 2(c). You can use freely the limit lims→0+ g(s) = 0, cf. Problem 2(b). The equations, all assuming t > 0, x > 0: etẋ = f(t)− xet(D1) ẋ = f(t) · (e− x)(D2) ẋ = f(t) · x3ex−2(D3) ẋ = f(t) · f(x)(D4) (Partial score can be awarded for general solutions.) 3 Problem 4 counts 2/3 of the hand-in For each real constant h, define the matrix Ah =  −4h 2 0 0 2 0 −1 1 0 2 0 −h 8 1 0 0 . Furthermore, let 1 denote the vector  1 1 1 1 . a) Is there any h and real q such that  1/2 0 0 −1 −4 0 0 1 17 7 −14 −6 16 0 −14 −4  −1 = qAh? b) Without calculating any cofactors in part b): (i) Why can you tell that the determinant |Ah| is zero when h ∈ {−4, 0}? (ii) Is it possible without further calculations to say that there is some h that yield(s) infinitely many solutions to the equation system Ahx = h1? c) For what values of h will the equation system Ahx = h1 have precisely one solution? d) Select either 2&3 or 1&4 (feel free to choose the one you think makes your task easier), and calculate two elements of the solution x of the equation system Ahx = h1, without using Gaussian elimination: Either you calculate x2 and x3, or you calculate x1 and x4. e) Invert A−8. 4

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