On-line Math 21

4.5  Curve Sketching

Example 4 f(x) = x^1/3 .

Solution

   Intercepts

Here the only intercept is at (0,0) .

   Increasing/decreasing, and critical points

Since

f¢(x) = 1 3 x-2/3,

f¢(x) is always positive.

   Concavity, and inflection points

Again, since

f¢¢(x) = - 2 9 x-5/3,

f¢¢(x) is positive when x is positive, negative when x is negative. Note that f¢(0) is undefined, and more specifically

lim x® 0  f¢(x) = ¥.

This information we just found is not quite as empty as it may seem. By saying there are no critical points definitely says that the graph does not level off, and since f(x) is never 0 except at the origin, you know it never crosses the horizontal axis again.

   Draw the graph

Here the drawing is a bit simpler, you don't need the several stages of plotting various information. Instead, start at 0 and maintain the proper slope and concavity.

Copyright (c) 2000 by David L. Johnson.