Second Derivative Test

We omit the proof and give a simple intuitive argument instead. (See Figure 3.5.8.) Since f'(c) = 0, the curve is horizontal at c. If f"(c) is negative the slope is decreasing. This means that the curve climbs up until it levels off at c and then falls down, so it has a maximum at c. On the other hand, if f"(c) is positive, the slope is increasing, so the curve falls down until it reaches a minimum at c and then climbs up. This argument makes it easy to remember which way the inequalities go in the test.

Figure 3.5.8

The Second Derivative Test fails when f"(c) = 0 and when f"(c) does not exist. When the Second Derivative Test fails any of the following things can still happen:

(1)    f has a maximum at x = c.

(2)    f has a minimum at x = c.

(3)    f has neither a maximum nor a minimum at x = c.

In most maximum and minimum problems, there is only one critical point except for the endpoints of the interval. We develop a method for finding the maximum and minimum in that case.