Problem

Consider a function f which is defined for all real numbers except x = 1, and assume f has the following properties.

1. f(x) = 0 if and only if x = 0
   
2. lim _{x —>–¥} f(x) = 0
   
3. lim _{x —>+¥} f(x) = 0
   
4. lim _{x —>1} f(x) = ¥
   
5. f '(x) > 0 when –1 < x < 1
   
6. f '(x) < 0 when x < – 1 or x > 1
   
7. f ''(x) > 0 when –2 < x < 1 or x > 1
   
8. f ''(x) < 0 when x < –2.

Draw a possible graph of f.

Solution

First, we translate into English the information that each property gives us.

1 states that the graph of f crosses the x axis only once, and that this occurs when x = 0.

2, 3, and 4 give us asymptotes of the function: there is a horizontal asymptote at y = 0 and a vertical asymptote at x = 1. Furthermore, since f only changes sign once, at x = 0 (by property 1), combining this with the fact that f gets arbitrarily (positively) large at x = 1, we must have f positive for x > 0. Since f changes sign at x = 0, this means that f is negative for x < 0.

5 tells us where f is increasing, and 6 tells us where f is decreasing. We can also put these two properties together to gain more information about f: since f ' changes from negative to positive at x = – 1, f must have a local minimum at that point, and since f ' changes from positive to negative at x = 1, f must have a local maximum at that point. The confirms what we already know, since we are told lim _{x —>1} f(x) = ¥.

7 tells us where f is concave up, and 8 tells us where f is concave down.

Putting all this information together, here is one possible graph of f.