In the previous section, we discussed the second derivative test and worked a max-min problem. Today we will also make use of the first and second derivative test, but first we need some new definitions.
A function is concave up on an interval if there is a local minimum on the interval. A function is concave down on an interval if there is a local maximum on the interval. We also have a new way to classify zeroes of f''. A point, (a,f(a)), is an inflection point if the zero of f''(a) has opposite signs to the left and right of it. In other words if there is a local minumum on one side of f''(a) and a local maximum on the other, the (a,f(a)) is a an inflection point. The inflection point is where f(x) changes concavity.
Let's practice some curve sketching with what we know so far. Suppose we had the function, . It's difficult to draw the graph of f(x) just by looking at it. We'll have to do some analysis. Finding the first derivative and second derivative for a start would help. and . Let's start with the first derivative. The values of x that make f' zero are x = -4 and x = 1. The first derivative test tells us to look at the sign of f' to find the local minumum or local maximum. Let's try to check the sign around these values. To the left of x = -4, f'(x) takes on positive values. In between x = -4 and x = 1, f'(x) takes on negative values. We already have a change in sign. If f'(c) > 0 on one side of c and f'(c) < 0 on the other then we have a local maximum a (c,f(c)). A local maximum tells us that the graph of f(x) will be concave down on that interval. Continuing to the the right of x = 1, f'(x) takes on positive values. We have another change in sign. To the left of x = 1, f'(x) < 0 and to the right of 1, f'(x) > 0. On an interval around c, where f' is less than zero to the left of c and f' is positive to the right of c, we have a local minumum. A local minimum on an interval tells us it concave up.
We already have a good deal of information about f(x). Let's see if the second derivative can give us any more facts about the behavior of f(x). . The value that makes f''(x) zero is -3/2. To the left of x = -3/2, we have a local maximum, and to the right of x = -3/2, f(x) has a local minimum. Since f''(-3/2) has opposite signs to the left and to the right, there is an inflection point at x = -3/2 which is where f(x) starts becoming concave up from being concave down.
We have a pretty good idea of what f(x) looks like. We can now plot what we know on a graph which is shown below.