contour-1 - 1 Solution to Problem 5 on 2008 Midterm 1 a. A...

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1 Solution to Problem 5 on 2008 Midterm 1 a. A local minimum of f is a point such that if we move away in any direction, the value of f increases. Such a point should satisfy the following properties: A local minimum should lie inside of a closed level curve. To see why this is the case, look at, for example, point E , which lies in a region which is not bounded by a closed curve. If we move to the right, we don’t know what happens to the value of the function, so we certainly can’t conclude that there’s a local minimum there. The closed curve containing a local minimum should not contain any level curve corresponding to a smaller value. To see why this is the case, look at point C . Point C is contained in a curve for which f ( x,y ) = - 2, but this curve contains a curve on which f ( x,y ) = - 3. This means that the value of f ( x,y ) at C is somewhere between - 2 and - 3, but if we move down just a little bit, we get to the level curve on which f ( x,y ) takes on the smaller value - 3. So there is not a local minimum at C . There are only 4 points that meet these criteria: A,D,F , and H . Let’s look in particular at point H . If we start at H and move (in any direction) to the nearest level curve, the value of the function
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contour-1 - 1 Solution to Problem 5 on 2008 Midterm 1 a. A...

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