Function Analysis: Increasing, Decreasing, and Concavity, Slides of Calculus for Engineers

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Lesson 4

Analysis of Functions I:

Increasing, Decreasing

and Concavity

OBJECTIVES:

• to define increasing and decreasing functions;
• to define concavity and direction of bending that is

concave upward or concave downward; and

• to determine the point of inflection.

The following definition, which is illustrated in Figure

4. 1. 2 , expresses these intuitive ideas precisely.

CONCAVITY Although the sign of the derivative of f reveals where the graph of f is increasing or decreasing , it does not reveal the direction of the curvature. Figure 4.1.8 suggests two ways to characterize the concavity of a differentiable f on an open interval:

• f is concave up on an open interval if its tangent lines have increasing slopes on that interval and is concave down if they have decreasing slopes.
• f is concave up on an open interval if its graph lies above the its tangent line and concave down if it lies below its tangent lines.

Formal definition of the “concave up” and “concave

down”.

Since the slopes of the tangent lines to the graph of a differentiable function f are the values of its derivative f’, it follows from Theorem 4. 1. 2 (applied to f’ rather than f ) that f’ will be increasing on intervals where f’’ is positive and that f’ will be decreasing on intervals where f’’ is negative. Thus we have the following theorem.