Applications of Differentiation: How Derivatives Affect the Shape of a Graph

Vocabulary

monotonicity, Increasing/Decreasing Test, First Derivative Test, concavity, Concavity Test, Second Derivative Test, inflection point

Objectives

• 1. Identify intervals on which the graph of a function is increasing or decreasing.
• 2. Identify intervals on which the graph of a function is concave upward or concave downward.
• 3. Classify critical numbers as local extrema using the first or second derivative tests.

Lecture Outline

Monotonicity

Definition of Increasing and Decreasing

We say that a function f is increasing on an interval I if whenever for in I. Similarly, we say that a function f is decreasing on an interval I if whenever for in I.

Note that we must always consider the graph of the function from left to right when determining the intervals over which the function increases or decreases.

Increasing/Decreasing Test

If the first derivative of a function is positive on an interval, then the function is increasing on that interval; if the the first derivative of a function is negative on an interval, then the function is decreasing on that interval.

The Increasing/Decreasing Test can be proven using the Mean Value Theorem.

Example

Determine the intervals over which is increasing or decreasing.

Solution:

We define f, compute its derivative, and then determine the sign of the derivative for all values of x.

> f:=x->2*x^3-3*x^2-12*x+7;

> fprime:=D(f)(x);

> solve(fprime>0,x);

The function is increasing on the intervals ( ) and (2, ).

> solve(fprime<0,x);

The function is decreasing on the interval (-1,2).

Example

Determine the intervals on which is increasing or decreasing.

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We can use information about the first derivative of f to classify the critical numbers of f as local maxima, local minima, or neither.

First Derivative Test

Suppose that c is a critical number of a continuous function f.

• 1. If f ' changes from positive to negative across c, then f has a local maximum at c.
• 2. If f ' changes from negative to positive across c, then f has a local minimum at c.
• 3. If f ' does not change sign across c, then f has neither a local maximum nor a local minimum at c.

Remark: The First Derivative Test will always provide a conclusive classification of any critical number of f as a local maximum, a local minimum, or neither.

Example

Find the local extrema of .

Solution:

We first compute f ' (x), then find the critical numbers, and finally classify the critical numbers using the First Derivative Test.

> f:=x->x^3-3*x+4;

In Maple, we can determine the critical numbers by examining the graph of f ' (x). The critical numbers occur where the graph of f ' (x) crosses the x-axis. The graph will also be useful when we classify the critical numbers since it shows where f ' (x) is positive or negative.

> plot(D(f)(x),x=-2..2);

There are critical numbers at -1 and 1. Notice that f ' changes from positive to negative across -1, so f has a local maximum at -1. Since f ' changes from negative to positive across 1, f has a local minimum at 1.

Concavity

Definition of Concavity

We say that the graph of a function f is concave upward on an interval I if the graph lies above all its tangents on I. The graph of a function f is concave downard on an interval I if the graph lies below all its tangents on I.

Concavity of a function on an interval I can be assessed by determining the sign of f '' (x) on I, as the following test states.

Concavity Test

• (a) If f ''(x) > 0 for all x in an interval I, then the graph of f is concave upward on I.
• (b) If f ''(x) < 0 for all x in an interval I, then the graph of f is concave downward on I.

The Concavity Test also can be proven using the Mean Value Theorem.

Inflection Points

We say that f has an inflection point at c if the concavity of f changes across c.

Question: If f ''(c)=0, does f have an inflection point at c?

We use the notion of concavity to develop another test (the Second Derivative Test) for classifying the critical numbers of a function f. The Second Derivative Test has the advantage that often it is easier to compute f ''(c) and determine its sign than it is to determine where f ' is positive or negative. Hence it is often possible to save time by applying the Second Deriative Test rather than the First Derivative Test in classifying the critical numbers of f. Unfortunately, the Second Derivative Test is not always conclusive. In those cases, we will have to rely on the First Derivative Test.

Second Derivative Test

Suppose f ' ' (x) is continuous near c.

• (a) If f ' (c) = 0 and f ' ' (c) > 0, then f has a local minimum at c.
• (b) If f ' (c) = 0 and f ' ' (c) < 0, then f has a local maximum at c.

Example

Find the local extrema of using the second derivative test.

Solution:

Note that since f is a polynomial function, its second derivative is continuous everywhere and the Second Derivative Test is applicable.

We determine the critical numbers of f.

> f:=x->3*x^5-5*x^3+3;

> solve(D(f)(x),x);

We define g as the second derivative of f.

> g:=x->D(D(f))(x);

Determine the sign of the second derivative at each critical number.

> g(0),g(1),g(-1);

The Second Derivative Test is inconclusive in classifiying the critical number 0. We do, however, find that f has a local minimum at 1 and a local maximum at -1.

We now plot the graph of the first derivative of f and use the First Derivative Test to classify the critical number 0 of f.

> plot(D(f)(x),x=-2..2,y=-5..5);

Since the sign of f ' does not change across 0, f has neither a local maximum nor a local minimum at 0.

Exercises

For each of the following functions, find the critical numbers and use either the first or second derivative test to classify them as local maxima or minima.

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2. , x not equal to 0.

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4. , .

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