Example 1 (Continued): Change of Direction

y = x³ + x - 1.

We have

dy/dx is always positive, while d²y/dx² = 0 at x = 0. We make a table of values for y and its first two derivatives at x = 0 and at a point to the right and left side of 0.

With the aid of Theorems 1-3, we can draw the following conclusions:

(a)    dy/dx > 0 and the curve is increasing for all x.

(b)    d²y/dx² < 0 for x < 0; concave downward.

(c)    d²y/dx² > 0 for x > 0; concave upward.

At the point x = 0, the curve changes from concave downward to concave upward. This is called a point of inflection.

To sketch the curve we first plot the three values of y shown in the table, then sketch the slope at these points as shown in Figure 3.7.7, then fill in a smooth curve, which is concave downward or upward as required.

Figure 3.7.7