MAT.126 3.3-3.4

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3.4 Concavity and the Second Derivative Test

Apply the Second Derivative Test to find relative extrema of a function

Theorem 3.9 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.

Find any points of inflection of the graph of a function

Theorem 3.8 Points of Inflection

If (c,f(c)) is a point of inflection of the graph of f, then either f''(c) = 0 or f'' does not exist at x = c.

Note: The converse is not, in general, true. For example, consider f(x) = x^4 at c = 0.

Let f be a function that is continuous on an open interval and let c be a point in the interval. If the graph of f has a tangent line at this point (c,f(c)), then this point is a point of inflection of the graph of f if the concavity of f changes from upward to downward (or downward to upward) at the point.

This definition is slightly more restrictive than is sometimes used in that it requires a tangent line to exist at c.

Determine intervals on which a function is concave upward or concave downward

Use x-values where f''(x)=0 or f'' does not exist to determine test intervals

Theorem 3.7 Test for Concavity

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

Understand concavity graphically

Definition

Let f be differentiable on an open interval I. The graph of f is concave upward on I if f' is increasing on the interval and concave downward on I if f' is decreasing on the interval.

3.3 Increasing and Decreasing Functions and the First Derivative Test

Apply the First Derivative Test to find relative extrema of a function

Theorem 3.6 The First Derivative Test

Let c be a critical number of a function f that is continuous on an open interval I containing c. If f is differentiable on the interval, except possibly at c, then f(c) can be classified as follows.

Determine intervals on which a function is increasing or decreasing

Guidelines

Guidelines For Finding Intervals On Which A Function Is Increasing Or Decreasing

Let f be continuous on the interval (a,b). To find the open intervals on which f is increasing or decreasing, use the following steps.

These guidelines are also valid if the interval (a,b) is replaced by an interval of the form (-infinity,b), (a,infinity), or (-infinity,infinity).

Theorem 3.5 Test for Increasing and Decreasing Functions

Let f be a function that is continuous on the closed interval [a,b] and differentiable on the open interval (a,b).

Definitions

A function f is increasing on an interval if for any two numbers x_1 and x_2 in the interval, x_1 < x_2 implies f(x_1) < f(x_2).

A function f is decreasing on an interval if for any two numbers x_1 and x_2 in the interval, x_1 < x_2 implies f(x_1) > f(x_2).