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Concavity and the Second Derivative Test

Definition of Concavity

If f(x) is a differentiable function on an open interval (a, b) and L is a tangent line to y = f(x), then the graph of f is concave upward on (a, b) when it lies above all of its tangents or concave downward on (a, b) when it lies below all of its tangents.

Concavity Test

Let f be a function whose second derivative exists on an open interval I.

  1. If f'' > 0 on I, the graph of f is concave upward in I.
  2. If f''< 0 on I, the graph of f is concave downward in I.

View the presentation designed to provide an intuitive approach to concavity and inflection points.

Points of Inflection

A point P on the curve y = f(x) is called an inflection point if f is continuous there and the curve changes from concave upward to concave downward or from concave downward to concave upward at P. View the presentations below that target concavity of a function and inflection points.

Second Derivative Test for Local Extrema

A consequence of the Concavity Test is the Second Derivative Test.

Let f be a function such that f'(c) = 0 and the second derivative of f exists on an open interval containing c.

  1. If f''(c) > 0, then f has a relative minimum at (c, f(c)).
  2. If f''(c) < 0, then f has a relative maximum at (c, f(c)).

If f''(c) = 0, the test fails, i.e., f may have a relative maximum or a relative minimum, or neither.

View the image and the presentation below illustrating how the Second Derivative Test is applied to finding relative and absolute extrema.

investigating local and absolute max/min instructions

A review of how the key features of a function are determined using calculus techniques is presented below.

Characteristics of Graphs of f(x), f'(x), and f''(x)

Many of the concepts presented in this module offer evidence of how the graphs of f(x), f '(x), and f ''(x) are interrelated. It is the derivative concept that offers the most enlightening view of the behavior of each of these functions. The table provides a summary of the relationships that exist among the graphs of f(x), f '(x), and f ''(x).

of f(x), f '(x), and f ''(x).

Concavity and the Second Derivative Test Practice

Concavity and the Second Derivative Test: Even More Problems

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of concavity and the Second Derivative Test.

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