Sketching a FunctionDate: 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. |
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