AP Calculus - Minimums and Maximums
Date: 12/23/98 at 23:54:01 From: Guppy Subject: AP Calculus The derivative of a function f is given by: f'(x) = (x^3-2x)(cos x) for 0 <= x <= 2 (A) Find the x-coordinate of the relative minimum of f(x). Show the analysis that leads to your conclusion. (B) Find the x-coordinate of each point of inflection on the graph of f(x). Justify your answer. (C) Find the x-coordinate of the point at which f(x) attains an absolute maximum. Justify your answer.
Date: 12/24/98 at 08:07:18 From: Doctor Jerry Subject: Re: AP Calculus Hi, Perhaps I can give a few hints so that you can finish these problems. (A) f'(x) = 0 at two points within (0,2), both easy to find. f' goes from negative to positive to negative. This will tell you about the relative minimum. (B) You can calculate f'', graph it to see that it has two zeros in (0,2), and use your calculator to determine these zeros. f'' changes sign at each zero, so they are points of inflection. (C) I'm not certain about this question and may be missing something. We don't know f and so can't compare f at 0 with f at the relative maximum in (0,2). The only thing that occurs to me is to go ahead and integrate f'. The constant doesn't matter because it just translates the graph and does not affect the x at which the absolute maximum occurs. I did this and found the absolute maximum to be at 0. - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/
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