Solutions to Tricky Problems

Solutions to Tricky Problems - Solutions to Tricky problems...

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1. The answer is (e). The curves meet in three places, ( - 2 , 4) , (0 , 0) and (3 , 9). The region between the curves is made up of a pair of leaf-shaped regions touching at the point (0 , 0). The x -intervals for the two regions are [ - 2 , 0] and [0 , 3]. In the first region, y goes from x 2 to x 3 - 6 x , while in the second region it is the other way around. Therefore, (c) would be right except that I forgot to change the x -limits on the second region; the correct answer is therefore (e). 2. The answer is (b). In (a) there is nothing distinguishing a maximum from a minimum. In (c) it is possible that the minimum is elsewhere on the boundary, for example at the origin. In (d) there is not a minimum in the interior, but it could be on one of the other boundary segments. To see that it is (b) and not (e), notice that if f x and f y are both greater than zero, then there can be no maximum on the interior (because the gradient does not vanish), on the interior of the x -axis segment (because f x > 0), on the interior of the y -axis (because f y > 0), or at the origin (because D u > 0 for any u pointing into the triangle). Any continuous function on a closed bounded set has a maximum (see page
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Solutions to Tricky Problems - Solutions to Tricky problems...

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