Objectives: In this tutorial, we define what is meant by a function that is concave upward or concave downward on some interval. Two geometrical conditions and two conditions using the derivative and second derivative are used to make this definition. Some examples of finding graphically where a given function is concave upward or concave downward are given. Inflection points are also defined. There are three quizzes that test the relationships between the graph of a function and information about its derivatives. After working through these materials, the student should be able

• to recognize graphically when a function is concave upwards or concave downwards by looking at the graph of its second derivative;
• to recognize graphically the inflection points of a function;
• given the graphs of a function, its derivative and second derivative, to determine which graph is which;
• to determine information about a graph of a function given information about its derivative and second derivative.

Modules:

 Theorem. Suppose that f is a function such that the second derivative of f exists for all x in the open interval I. The following are equivalent statements. f ''(x) > 0 for all x in I. f ' is an increasing function on I. For each c in I, the tangent line to the graph of f at x = c lies below the graph of f. For each pair of distinct points, a and b, in I, the line segment between (a, f(a)) and (b, f(b)) lies above the graph of f. Definition. In this case, we say that f is concave upward on I.

• Example.

• Similarly, we have the following.

 Theorem. Suppose that f is a function such that the second derivative of f exists for all x in the open interval I. The following are equivalent statements. f ''(x) < 0 for all x in I. f ' is an decreasing function on I. For each c in I, the tangent line to the graph of f at x = c lies above the graph of f. For each pair of distinct points, a and b, in I, the line segment between (a, f(a)) and (b, f(b)) lies below the graph of f. Definition. In this case, we say that f is concave downward on I.

• The proof of these Theorems will be given in another module.

• Example.

• Using a graphing calculator to graph a function and its second derivative.

 Definition. Suppose that a is in the domain of the function f and suppose that there exists an open interval (c, d) containing a such that either f is concave upward on (c, a) and is concave downward on (a, d); or f is concave downward on (c, a) and is concave upward on (a, d). Then a is called an inflection point of f.

• A LiveMath notebook investigating functions, concavity and inflection points.

• Quiz that tests the ability to determine graphically information about a function and its derivatives.

• Quiz which tests the ability to determine information about the graph of a function from information about its derivatives.

• Quiz on determining which graph is the graph of a function, its derivative and its 2nd derivatives.