Sketching a Function

Date: 07/06/98 at 08:53:52
From: MatherWu
Subject: Sketching a function

Dr. Math:

I have question. Would you help me solve this problem? 

Sketch a function g that is increasing and concave up on 
(-infinity, 1), decreasing and concave down on (1, infinity), 
such that g(1) = 3.

Thanks.

Date: 07/07/98 at 18:18:58
From: Doctor Peterson
Subject: Re: Sketching a function

Hi, MatherWu. This problem is sort of like those kits that police 
artists use to sketch a criminal's face by putting together the right 
kind of nose, eyes, and so on. What you need is to put together the 
different pieces of the function based on the description.

Here are the standard pieces in our kit:

   Convave up, decreasing        Concave up, increasing

       *                                            *
        *                                          *
          *                                      *
             *                                *
                   *                    * 

   Concave down, increasing      Concave down, decreasing
 
                   *                    *   
             *                                *
          *                                      *
        *                                          *
       *                                            *

You want your function to be concave up and increasing everywhere to 
the left of x = 1, then concave down and decreasing everywhere to the 
right of x = 1, with y = 3 at x = 1. So we can put these pictures 
together and label the place where they join:

   Concave up, increasing     Concave down, decreasing
                 |         |
                 |         |(1,3)
                 |         * 
                 |        *|     *
                 |      *  |        *
                 |   *     |          *
               * |         |           *
                 |         |
   --------------+---------+------------------
                 0         1

You probably notice that this function doesn't have a continuous slope 
- it has a "break" at x = 1, since it has to have a positive slope to 
the left and a negative slope to the right. That's not typical of the 
functions we usually work with, but it's perfectly legal for a 
function. If it had been concave down on the left, it could have 
looked like a parabola pointing down, with no "cusp" (point) at the 
top.

- Doctor Peterson, The Math Forum
Check out our web site! http://mathforum.org/dr.math/   

Date: 07/07/98 at 23:39:22
From: Anonymous
Subject: Re: Sketching a function

Dear Doctor Peterson, 
I received your sketching a function.
Thanks.