Derivative Test

Many applications of calculus require us to deduce facts about a function f from the information concerning its derivatives. Since f ‘ (x) represents the slope of the curve y = f(x) at the point (x, f(x)), it tells us the direction in which the curve proceeds at each point.

Increasing / Decreasing Test

1. If f(x0 > 0 on an interval, then f is increasing on that interval.
2. If f(x0 < 0 on an interval, then f is decreasing on that interval.

Example:

Find where the function f(x) = x³ – 12x + 1 is increasing and where it is decreasing.

Solution:

Step 1: Find the derivative of f

f ‘(x) = 3x² – 12 = 3(x² – 4) = 3(x –2) (x + 2)

Step 2: Set f ‘(x) = 0 to get the critical numbers

f ‘(x) = 3(x –2) (x + 2) = 0
x = 2, –2

Step 3: Set up intervals whose endpoints are the critical numbers and determine the sign of f ‘(x) for each of the intervals. Use the increasing/decreasing test to determine whether f(x) is increasing or decreasing for each interval.

The First Derivative Test

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

1. If f ‘ changes from positive to negative at c, then f has a local maximum at c.
2. If f ‘ changes from negative to positive at c, then f has a local minimum at c.
3. If f ‘ does not change sign at c (f ‘ is positive at both sides of c or f ‘ is negative on both sides), then f has no local maximum or minimum at c.

Example:

Find the local maximum and minimum values of the function f(x) = x⁴ – 2x² + 3

Solution:

Step 1: Find the derivative of f

f ‘(x) = 4x³ – 4x = 4x(x² –1) = 4x(x –1)(x +1)

Step 2: Set f ‘(x) = 0 to get the critical numbers

f ‘(x) = 4x(x –1)(x +1) = 0
x = –1, 0, 1

Step 3: Set up intervals whose endpoints are the critical numbers and determine the sign of f ‘(x) for each of the intervals.

Step 4: Use the first derivative test to find the local maximum and minimum values.

f ‘(x) goes from negative to positive at x = –1, the First Derivative Test tells us that there is a local minimum at x = –1.
f (–1) = 2 is the local minimum value.

f ‘(x) goes from positive to negative at x = 0, the First Derivative Test tells us that there is a local maximum at x = 0.
f (0) = 3 is the local maximum value.

f ‘(x) goes from negative to positive at x = 1, the First Derivative Test tells us that there is a local minimum at x = 1.
f (1) = 2 is the local minimum value.

The Second Derivative Test

We can also use the Second Derivative Test to determine maximum or minimum values.

The Second Derivative Test

Suppose f ‘’ is continuous near c,

1. If f ‘(c) = 0 and f ‘’(c) > 0, then f has a local minimum at c.
2. If f ‘(c) = 0 and f ‘’(c) < 0, then f has a local maximum at c.

Example:

Use the Second Derivative Test to find the local maximum and minimum values of the function f(x) = x⁴ – 2x² + 3

Solution:

Step 1: Find the derivative of f

f ‘(x) = 4x³ – 4x = 4x(x² –1) = 4x(x –1)(x +1)

Step 2: Set f ‘(x) = 0 to get the critical numbers

f ‘(x) = 4x(x –1)(x +1) = 0
x = –1, 0, 1

Step 3: Find the second derivative

f ‘’(x) = 12x² – 4

Step 4: Evaluate f ‘’at the critical numbers

f ‘’(–1) = 8 > 0, so f (–1) = 2 is the local minimum value.
f ‘’(0) = – 4 < 0, so f (0) = 2 is the local maximum value.
f ‘’(1) = 8 > 0, so f (1) = 2 is the local minimum value.

Videos

Maxima and Minima
Finding relative maxima and minima of a function can be done by looking at a graph of the function. A relative maximum is a point that is higher than the points directly beside it on both sides, and a relative minimum is a point that is lower than the points directly beside it on both sides. Relative maxima and minima are important points in curve sketching, and they can be found by either the first or the second derivative test.

The First Derivative Test for Relative Maximum and Minimum
The first derivative test is a way to find if a critical point of a continuous function is a relative minimum or maximum. Simply, if the first derivative is negative to the left of the critical point, and positive to the right of it, it is a relative minimum. If the first derivative test finds the first derivative is positive to the left of the critical point, and negative to the right of it, the critical point is a relative maximum.

Concavity and Inflection Points
Inflection points are points on the graph where the concavity changes. A positive second derivative means a function is concave up, and a negative second derivative means the function is concave down. These inflection points are places where the second derivative is zero, and the function changes from concave up to concave down or vice versa.

The Second Derivative Test for Relative Maximum and Minimum
The second derivative test is useful when trying to find a relative maximum or minimum if a function has a first derivative that is zero at a certain point. Since the first derivative test fails at this point, the point is an inflection point. The second derivative test relies on the sign of the second derivative at that point. If it is positive, the point is a relative minimum, and if it is negative, the point is a relative maximum.

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