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/