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Math1210 Weekly Assignment 7 (Local and extremal max/min - Practical problems 3.3, 3.4) Spring, 2016 Uid number: instructor: name: Instructions: Please show all of your work as partial credit will be given where appropriate, and there may be no credit given for problems where there is no work shown. All answers should be completely simplified, unless otherwise stated. No calculators or electronics of any kind are allowed. (Total of 45 points possible.) 1. (10 points) Solve the following problems using derivatives. Make sure you set the right range for the variables. (a) Find two numbers whose product is −8 and the sum of whose squares is a minimum. (b) For what non-negative number does the square root exceed eight times the number by the largest amount? 2. (25 points) Find the critical points for each of the following functions in the indicated interval (if not stated, the interval is the whole real line). Then, use a test of your choice to decide which critical points give a local maximum or minimum value. Finally, identify the global maximum and minimum in the indicated interval, if they exists. (a) f (x) = x4 + x2 + 3 (b) f (x) = x x2 + 4 (c) f (x) = cos2 2θ 0≤θ≤π Page 2 √ (d) f (x) = 6 x − 4x (e) f (x) = x−1 x2 − 4 0≤x≤4 x≥0 Page 3 3. (10 points) A rectangle has to be inscribed in a semicircle of radius r, as shown in the figure below. What are the dimensions of the rectangle if we want the area to be maximal? Page 4