Example 1: Reading a Graph

Click here to open an associated Mathcad worksheet:

Problem. In the figure below, give the intervals over which the function is:

1. increasing
2. decreasing
3. neither increasing nor decreasing
4. concave up
5. concave down
6. neither concave up nor concave down (i.e. linear)

Solution.

1. x < a,  b < x < 0,  c < x < d,  f < x.
2. a < x < b,  d < x < f.
3. 0    x    c. Notice that at the points  a, b, d, and  f  the function is likewise neither increasing nor decreasing. We call these points extrema of the function.
4. a < x < 0,  e < x.
5. c < x < e. A point like  e  where the concavity changes from down to up (or vice versa) is called an inflection point of the function.
6. x    a,  0    x    c.