stationary_points.20100216.4b7b533cd07728.83001590

stationary_points.20100216.4b7b533cd07728.83001590 - Study...

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Study sheet — Stationary Points Harry Dankowicz Mechanical Science and Engineering University of Illinois at Urbana-Champaign danko@uiuc.edu Objective: To investigate the conditions for a local maximum or minimum of a function. Observation 1 Consider the function f ( x, y )=5 x y . As the rate of change of f with respect to either of its arguments is constant and nonzero, f either increases or decreases under any variations in either of its arguments and remains constant if and only if x and y vary along a line in R 2 with slope 5. Each such line may be characterized by the intersection with the y -axis, say y , such that f ( x, y )= y along this entire line. It follows that the value of f decreases as y increases. Now restrict attention to those points in R 2 that live on the unit circle centered on the origin, i.e., such that x 2 + y 2 =1 . (1) We know that f is constant along each member of the family of straight lines in R 2 with slope 5. Only a subset of these lines intersect the unit circle. Of these, the one with the smallest value of y corresponds to the largest value of f along the circle. Similarly, the one with the largest value of y corresponds to the smallest value of f along the circle. Each of these two lines are tangent to the circle at the corresponding points of intersection. The corresponding points of intersection must therefore lie on a straight line through the origin with slope 1 / 5. It follows that f μ 1 26 , 5 26 = 10 26 (2) is the largest value of f along the circle and f μ 1 26 , 5 26 = 10 26 (3) is the smallest value of f along the circle.
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stationary_points.20100216.4b7b533cd07728.83001590 - Study...

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