As preliminary work we name our function g(x) and compute g'(x), g''(x), and a few values for g:
x | |
g(x) = | |
(x^{2} - 4) |
1*(x^{2} - 4) - x*(2x) | - (x^{2} + 4) | ||
g'(x) = | | = | |
(x^{2} - 4)^{2} | (x^{2} - 4)^{2} |
(-2x)*(x^{2} - 4)^{2} - ( -(x^{2} + 4) )*( 2(x^{2} - 4)(2x) ) | |||
g''(x) = | | = | |
(x^{2} - 4)^{4} | |||
(-2x)(x^{2} - 4) +(x^{2} + 4)(4x) | (2x)(x^{2} + 12) | ||
| = | | |
(x^{2} - 4)^{3} | (x^{2} - 4)^{3} |
We now go down the list:
The only obvious value of the function is g(0) = 0. So at this point we have:
The concavity of the graph is governed by the sign of g'(x). The numerator of g'(x) is positive if x is positive and negative if x is negative. It is zero at x = 0 and changes sign there. The denominator is positive for
Range of x-values: | < - 2 | -2 | -2 to 0 | 0 | 0 to 2 | 2 | > 2 |
---|---|---|---|---|---|---|---|
Sign of Numerator: | - | - | - | 0 | + | + | + |
Sign of Denominator: | + | 0 | - | - | - | 0 | + |
Sign of g''(x): | - | DNE | + | 0 | - | DNE | + |
Thus the graph of g(x) is concave down for x < -2, concave up for -2 < x < 0, and so on. Since the concavity changes at x = 0, this is an inflection point.
Finally, we note that g(x) itself is negative for large negative numbers and positive for large positive numbers, which tells us on which side the graph approaches x = 0.
Putting this all together, the graph starts below x =0 and is concave down so must go to - to the left of x = -2. To the right of x = -2 the graph is concave up, so it must go to to the right of x = -2. By symmetry, or continuing with the information given by the second derivative, the graph goes to - to the left of x = 2 and to - to the right. With no critical points other than the places where g is not defined, there are no bumps or other features. We get the following sketch which is displayed next to a computer-generated graph.
©
CalculusQuest^{TM}
Version 1
All rights reserved---1996
William A. Bogley
Robby Robson