In these lessons, we will learn
Related Topics: More Calculus Lessons
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.
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) = x^{3} – 12x + 1 is increasing and where it is decreasing.
Solution:
Step 1: Find the derivative of f
f ‘(x) = 3x^{2} – 12 = 3(x^{2} – 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.
Interval |
x – 2 |
x + 2 |
f ‘(x) |
f |
x < – 2 |
– |
– |
+ |
Increasing on |
–2 < x < 2 |
– |
+ |
– |
Decreasing on (–2, 2) |
x > 2 |
+ |
+ |
+ |
Increasing on |
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) = x^{4} – 2x^{2} + 3
Solution:
Step 1: Find the derivative of f
f ‘(x) = 4x^{3} – 4x = 4x(x^{2} –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.
Interval |
4x |
x –1 |
x + 1 |
f ‘(x) |
x < –1 |
– |
– |
– |
– |
–1 < x < 0 |
– |
– |
+ |
+ |
0 < x < 1 |
+ |
– |
+ |
– |
x > 1 |
+ |
+ |
+ |
+ |
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.
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) = x^{4} – 2x^{2} + 3
Solution:
Step 1: Find the derivative of f
f ‘(x) = 4x^{3} – 4x = 4x(x^{2} –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) = 12x^{2} – 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.
Relative Maxima and Minima (Local 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.
Increasing and Decreasing Functions
Determine the intervals for which a function is increasing and/or decreasing by using the first derivative.
This video provides an example of how to determine the intervals for which a function is concave up and concave down as well as how to determine points of inflection.
This video provides an example of how to use the second derivative test to determine relative extrema of a function.