FAX Management Kdan Cloud ΑΞ Share Tool + Topic 4 Questions Outlines Q Search Optimization Unconstrained optimization Constrained optimization (... TTO✓ Comment & Markup T Annotation Editor OCR OCR Convert Fax Search 66. Determine the extrema of the following functions and apply the second-derivative test to decide on their nature. (a) f(x, y) = x²y+y³ - y (b) f(x,y) z² - 32 (c) f(x, y) = x²y - y²+3 67. Consider the function f(x, y) = x²y³. (a) Find all critical points of the function f. (b) Investigate the nature of the critical points using the second-derivative test. (c) Investigate the nature of the critical points using a sign chart for f. 68. A manufacturer produces two types of toys A and B. The production cost for a unit of toy A amounts to €70; the production cost for a unit of toy B is €80. The demand functions for both toys are given by the formulas q₁ = 160 - 2PA + PB and QB = 60+ PA - PB, where PA and PB denote the unit prices for a unit of toy A and B respectively and qд and qɛ denote the amounts demanded. Calculate the selling prices PA and pg that maximize the weekly profit. (It suffices to check that you have found a relative maximum.) ЧА 69. The number of units of a certain product, denoted by q, that can be produced daily depends on the daily available quantities QA and qB of two commodities A and B: q = 10-11 The selling price per unit of the product amounts to €90. Commodities A and B cost €10 and €40 respectively. Find the maximal profit that can be obtained daily. (Again, it suffices to find a relative maximum.) 70. The sum of three non-negative numbers is 1200. What is the maximal value of their product? 71. A box in the form of a cuboid must have a volume of 16 m³. The material for the top and the bottom costs €20 per m². The material for the sides costs €10 per m². What is the minimal cost for the box? 72. Find the dimensions of the rectangular box with a content of 1 m³ and with the smallest total surface (i.e., the surface of all six faces together). 195% Paco 21 1