MATH 402 Midterm: MATH 402 2005 Winter Test 1
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0 px(t) (1 + x(t)2) dt : x(0) = 1, x(b) = 5 + m(b 2)) . Find the slope m (a constant) corresponding to each extremal. (c) for each extremal in item (b), use second-order tests to support the strongest conclusions you can justify concerning optimality or non-optimality. Choose b > 0 and x: [0, b] r to minimize. 0 x(t)2 x(t)2 dt subject to the endpoint restrictions x(0) = 1 and x(b) =p4 + b2. Consider this isoperimetric problem on a xed interval, assuming > 0: min x (cid:26)z 2. 0 p1 + x(t)2 dt : x(0) = 0 = x(2), z 2 x(t) dt = (cid:27) . 0 (a) prove: if a minimizing arc exists, its graph must be an arc of an ellipse in the (t, x)-plane. (b) prove: when = 1, a minimizing arc does exist, and indeed its graph is an arc of a circle in the (t, x)-plane.
