s23 - area under the graph of f(x) is a constant, its...

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Solution to Problem 23 Congratulations to this week’s winner Kevin Bourrillion A correct solution was also received from Yan Fridman. One can solve the problem as a straightforward minimization problem in differential calculus. But Kevin Bourrillion’s sparkling solution requires no knowledge of calculus at all. Here ’tis. First note that a simple vertical shift of the graph allows one to assume that the graph of f(x) is positive throughout the interval in question. Also the condition on the concavity means that the graph of any tangent line is above the graph of f(x) throughout, except, of course, at the point of tangency, where they are equal. Next, note that since the
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Unformatted text preview: area under the graph of f(x) is a constant, its enough to minimize the area under the graph of T c (x) , the tangent line at the point c . This area is that of a trapezoid and as such is given by the formula (b - a) (T c (a) + T c (b))/2 . Since the tangent is a staight line, this is further equal to (b - a) T c ((a + b)/2) . By the observation above, the term T c ((a + b)/2) is greater than f((a + b)/2) for all values of c not equal to (a+b)/2 . Therefore, the area is minimized when c = (a + b)/2 , that is, for the tangent line at the midpoint....
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This note was uploaded on 03/01/2010 for the course MATH 301 taught by Professor Albertodelgado during the Spring '10 term at Bradley.

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s23 - area under the graph of f(x) is a constant, its...

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