On-line Math 21

4.5  Curve Sketching

Example 5 f(x) = x⁴+4x³+4x²-2 .

Solution

   Intercepts

Not every calculation can be equally easy for these functions. Typically, the problems you see in the texts have perhaps one, or two, of: intercepts, or critical points, or inflection points, easy to find. The others are usually difficult to evaluate. That is especially true with this problem. I do not want to find some 4-degree formula, like the quadratice formula, to solve it. Such a formula does exist, but you don't want to see it.

The only intercept we can easily find is the y -intercept at (0,-2) . I used Maple tro draw a graph, and saw that there are x -intercepts at about 1/2 and -5/2 (they are not exactly at those points, but close).

   Increasing/decreasing, and critical points

f¢(x) = 4x³+12x²+8x , so critical points occur when

0 = f¢(x) = 4x3+12x2+8x = 4x(x2+3x+2) = 4x(x+1)(x+2),

so the only places where f¢(x) = 0 are x = 0,-1,-2 . The values are: f(0) = -2 ,

f(-1) = (-1)4+4(-1)3+4(-1)2-2 = 1-4+4-2 = -1,

and

f(-2) = (-2)4+4(-2)3+4(-2)2-2 = 16-32+16-2 = -2,

or the critical points and values are (0,-2), (-1,-1) , and (-2,-2) .

In between, the sign of f¢(x) is given as follows:

f¢¢(x) = 12x²+24x+8 , so inflection points occur at solutions to

0 = f¢¢(x) = 12x2+24x+8 = 4(3x2+6x+2),

which has roots at

x = -1± 1 Ö3 ,

using again the quadratic formula, or, really, Maple. Finding the y -values of these points is a hassle, let's not.

Here is the number line indicating the sign of the second derivative:

Use this information to draw a fair representation of the graph.

• The first information you plot are the intercepts and endpoints; [Link to ex5-1.gif]
• Then plot the critical points (and values). I always mark it with a horizontal line to remind myself that it is a critical point. [Link to ex5-2.gif]
• Plot the inflection points. I mark them with an ``I'' to indicate that they are inflection points. [Link to ex5-3.gif]
• Then fill in the graph, connecting the dots and making sure that the graph you draw is increasing/decreasing where the number line indicates it should be, and concave-up or concave-down where the sign of the second derivative indicates. [Link to ex5-4.gif]

[Each of these items should trigger the appearance of a new drawing with that information added.]

Copyright (c) 2000 by David L. Johnson.