Chapter 8 - Understanding the Derivative
Section 8.2 - Using the Second Derivative
The first derivative allows us to define equilibrium points on the graph of a function, . By evaluating points to the left and right of the equilibrium point we can classify these points as either maximums or minimums and thus determine the concavity of the graph. Without having found the equilibrium points it is extremely difficult to determine the behavior of a function over an interval. The sign of the first derivative only tells us if a function is increasing or decreasing; however, a function can increase or decrease in two way. For example consider the graphs of the following two different functions.
In both cases the function is increasing and the first derivative is always positive; however each function increases in a different way i.e. one increases concave up and the other increases concave down. Using the first derivative only, we would have to know not only where its positive or negative but also how the first derivative is changing i.e. positive and increasing, negative and increasing etc.
For the first graph
is positive and increasing thus the graph of f(x) is increasing and concave up. For the second graph,
is also positive but is decreasing. Thus the graph of
is concave down. The process of looking only at the graph of the first derivative to understand how
behaves is an extremely abstract and difficult one. To quicken and simplify our work we can use the function’s second derivative to conclude where the graph is concave up or down. This information along with the fact that the derivative is either positive or negative over an interval will be enough to accurately determine a function’s behavior.
Recall that a property of a concave up part of a graph is that its slope or rate of change is always increasing.
On the left side the slope is negative; however, as x increases the slope gets less and less, -5, -3, -2, till it reaches 0, from where on it
increases to 1, 5, 6, etc. We can then conclude that the rate at which the slope is changing must be positive or the graph of the derivative is increasing. Since the derivative’s value is constantly increasing, then the rate of change of the derivative, given by
will be positive. Remember that positive rate of change implies that the function is increasing over that interval, while a negative rate of change implies the function decreases as x increases.
In a concave up graph the derivative is increasing, such that the second derivative over this interval will be positive. Working in reverse we arrive at an important conclusion. If the second derivative is positive over an interval, then the first derivative is increasing, implying that the graph of the original function,
is concave up. This is true because the rate of change of a concave up graph is always increasing.
The reverse is true for concave down graphs. If the second derivative is negative then the first derivative is decreasing, implying that the original functions graph is concave down over the interval. The following graph summarizes the conclusions:
Though all the information concerning the behavior of f(x) can be obtained from studying its derivative, we can quicken and confirm our sketches by looking at the functions second derivative. Without having found any equilibrium points we can accurately determine the behavior of
over an interval by using the signs of both the function’s first and second derivative simultaneously. Four possibilities may exist for the signs of the derivatives.
are positive over an interval. Therefore
is increasing and concave up.
is positive but
is negative. Thus
is increasing and concave down
are negative, in which case
is decreasing and concave down.
is negative but
is positive, thus
is decreasing and concave up.