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Exam 2 bonus (+10%) (from P. Yasskin and A. Belmonte) Upon moving to a new city, you want to find an apartment which is conveniently located relative to your school (point S in 2-dimensional space), your place of work (point W ) and the shopping mall (point M ). You want to minimize the sum of distances from your apartment A = (x, y) to the above ~ + |AW ~ | + |AM ~ |. Find and prove the geometric three points: f (x, y) = |AS| ~ AW ~ and AM ~ which characterizes the condition on the three vectors AS, minimum of f . You may assume that A, W and M form an acute triangle. Hints: ~ expressing your answer in terms of the 1. Compute the gradient of |AS| ~ In particular, how are their directions related and what can vector AS. you say about the magnitude of the gradient? 2. Write the condition for the critical point in the vector notation. ~ 3. The final answer is a condition on the angles between the vectors AS, ~ and AM ~ . AW 4. (hard, +10% extra) Discuss how the answer would change if one of the angles in the triangle SW M were large (how large exactly?) and why does the above proof fail (why differentiation fails to locate the critical point). 1