Study Sheet (4.1)
Definition of Increasing and Decreasing Functions
If y = f (x) for x in the interval a < x < b ,
• f is an increasing function, if
,
• f is a decreasing function, if
• f is a
.
function, if f (a) = f (b) .
Test for Increasing and Decreasing Functions
Let f be differentiable on the interval (a, b) ,
• If f ' ( x) > 0 for all x in (a, b) , then
on (a, b) .
• If f ' ( x) < 0 for all x in (a, b) , then
on (a, b) .
• If
for all x in (a, b) , then f is constant on (a, b) .
Definition of Critical Number
If f is defined at c , then c is a critical number of f if
.
Example: Find all critical numbers for each function over its domain.
(remember that a critical number must be in the domain of the original function.)
(a) f ( x) = x 2
1
(b) f ( x) =
x
(c) f ( x) = e x
Example: A function f has three critical numbers as shown below. Determine on which of the ntervals
f is increasing or decreasing.
f ' ++++ 0 +++++++ u − − − − − − − − − − − − − 0 ++++++++++++++
P1
P2
P3
Recall that “u” means that the derivative is undefined at this critical number.
Definitions
Let f be a function defined at c .
A. Local maximum
• A function f has a relative maximum f (c) at a point x = c if f gets no larger near c . That is, if
f ( x) ≤ f (c) for all x near c .
A. Local minimum
• A function f has a relative minimum f (c) at a point x = c if f gets no smaller near c . That is, if
f ( x) ≥ f (c) for all x near c .
[“near” means on either side of x = c or to one side of x = c if c is an endpoint.]
A. Absolute maximum
• A function f has an absolute maximum f (c) at a point x = c if f gets no larger anywhere in the
domain under consideration. That is, if f ( x) ≤ f (c) for all x in a domain.
A. Absolute minimum
• A function f has an absolute minimum f (c) at a point x = c if f gets no smaller anywhere in the
domain under consideration. That is, if f ( x) ≥ f (c) for all x in a domain.
Note: The phrase “in the domain” means that we are considering either (1) the entire domain of the
function or (2) some interval smaller than the domain of the function.
Example: Label the following points on the graph of the function below.
(a)
(b)
(c)
(d)
Local maximum points
Local minimum points
The point where the function has its absolute maximum value.
The point where the function has its absolute minimum value.
a
b
Example: A function f has three critical numbers as shown below. Determine at which of the
points f has local maximum and minimum.
f ' ++++ 0 +++++++ u − − − − − − − − − − − − − 0 ++++++++++++++
P1
P2
P3
Recall that “u” means that the derivative is undefined at this critical number.
A Major Result (Used to locate Local Extrema)
If a function f has local maximum or minimum at x = c , then c must be a critical number
of f .
Or in other words,
All local maxima and minima occur where the derivative is either zero or undefined.
The Extreme Value Theorem
Every continuous function on a closed interval has both an absolute maximum and an absolute
minimum value.
Fermat’s Theorem
If f has a local maximum or minimum at x = c , and if f ' (c) exists, then f ' (c) = 0 .
Study Sheet (4.2)
Rolle’s Theorem
If f (x) is a differentiable function over [a,b], and if f(a) = f(b) = 0, then there is at least one point c
between a and b such that f’(c)=0:
Mean Value Theorem for Derivatives
If f (x) is a differentiable function over [a,b], then at some point between a and b:
f (b) − f (a)
f ' (c ) =
b−a
Note: Differentiable implies that the function is also continuous
The Mean Value Theorem only applies over a closed interval.
The Mean Value Theorem says that at some point in the closed interval, the actual slope equals
the average slope.
⎡ π⎤
Example: Show the function f ( x) = cos x on ⎢0, ⎥ satisfies the hypothesis of the Mean ValueTheorem.
⎣ 3⎦
The function is continuous on [0,π/3] and differentiable on(0,π/3). Since f(0) = 1 and f(π/3) = 1/2, the
Mean Value Theorem guarantees a point c in the interval (0,π/3) for which
f (b) − f (a)
1/ 2 −1
c = 0.498
f ' (c ) =
= − sin c
b−a
π /3−0
Corollary: Increasing Functions, Decreasing Functions
Let f be continuous on [a,b] and differentiable on (a,b).
1. If f’ > 0 at each point of (a,b), then f increases on [a,b].
2. If f’ < 0 at each point of (a,b), then f decreases on [a,b].
Note: A function is increasing over an interval if the derivative is always
A function is decreasing over an interval if the derivative is always
.
.
Study Sheet (4.3)
We’ll investigate the relationships between derivatives and graphs in this section.
Recall: First Derivative Test for Local Extrema at a critical point c
1. If f ‘ changes sign from positive to negative at c, then f has a local
at c.
2. If f ‘ changes sign from negative to positive at c, then f has a local
at c.
3. If f ‘ changes does not change sign at c, then f has
First derivative:
minimum.
Concavity
The graph of a differentiable function y = f(x) is
a. concave up on an open interval I if y’ is increasing on I. (y’’>0)
b. concave down on an open interval I if y’ is decreasing on I. (y’’<0)
Second Derivative Test for Local Extrema at a critical point c
1. If f’(c) = 0 and f’’(c) < 0, then f has a local maximum at x = c.
2. If f’(c) = 0 and f’’(c) > 0, then f has a local minimum at x = c.
Second derivative:
.
Point of Inflection
A point where the graph of a function has a tangent line and where the concavity changes is called a
point of inflection.
2
Example: Sketch the graph y = x3 − 3 x 2 + 4 = (x + 1)(x − 2 )
Make a summary table:
x
5
4
3
2
1
0
-2
-1
0
-1
1
2
3
4
y
y'
y' '
Worksheet (4.1-4.3)
PART 1
1. Use the graph of f ( x) shown below with domain [-3,12] to find the following:
f (5 x)
4
3
2
1
-3
-2
0
-1 -1 0
1
2
3
4
5
6
7
8
9
10
11
12
-2
-3
-4
A. Find the values of x where f ʹ( x) = 0 .
B. Find the values of x where f ʹ( x) is undefined.
C. Find the inflection points of f ( x) .
D. Find all the local maximums and minimums of f ( x) .
E. Find all the global maximums and minimums of f ( x) .
2. Using the graph and your answers to the questions above, do the following:
A. Label all the critical points on a number line. Determine the sign of f ʹ( x) between each critical point. How
does the information on this number line help you determine which critical points correspond to local
maximums, minimums, or neither?
B. Label all the points on a number line where f ʹʹ( x) is either zero or undefined. Determine the sign of f ʹʹ( x)
between each marked point. How does the information on this number line help you determine which points
correspond to inflection points?
PART 2
1. In each case, sketch a graph of a continuous function with the given properties.
-A. f ʹ(−1) = 0 and f ʹ(3) = 0
f ʹ( x)
+
--
|
-1
|
3
+
--
f ʹʹ( x)
|
2
+
B. g ʹ(1) = 0 and g ʹ(4) is undefined
g ʹ( x)
--
--
|
1
|
4
--
+
g ʹʹ( x )
|
4
+
C. hʹ(−2) = 0 and hʹ(2) = 0
hʹ(0) is undefined
hʹ( x)
-|
-2
+
--
hʹʹ( x)
--
|
0
|
2
-|
0
3. Determine i) critical points, ii) local extrema, iii) inflection points, and iv) intervals where f ( x) is concave up
or down. Include an accurate graph that illustrates these features.
8 x − 16
a. f ( x) =
x2
b. f (x) =
ex
1− e x