Aim: How can we use the first derivative to determine the relative extrema?
Objectives: to apply the first derivative to locate relative extrema when given a graph.
Grouping: students are given opportunity to work in cooperative setting during this class. Time is
purposely set aside for students to work collaboratively on at least one set of exercise problems. During
this time, students are encouraged to explain concepts and materials and solutions to each other.
Furthermore, students get to present their solutions on the board.
Differentiated instruction: all students are held to the highest standard. However the students that are
not performing well are grouped with students that are excelling to serve as study partners. Teacher will
informally assess their understanding by asking them questions to ensure that they are keeping up.
Assessment: Class is given multiple occasions to work individually and in groups. Teacher circulates
the room to assist and also assess student understanding. Furthermore, students are encouraged to
explain work to each other. This is another opportunity for teacher to assess student understanding.
Lastly, when lesson is finished, students are grouped for class work, questions that are designed as exit
slip problems.
Homework Review (10 minutes): Students are assigned problems to present on board. Students will
also answer questions. Teacher can go over a problem if no one in class could explain it
Do Now: (2007 AB-6): Let f be the function defined by f ( x)  k x  ln x for x  0 , where k is a
positive constant. [8 – 10 min]
a) Find f '( x)
1
1
k
1
ANS: f '( x)  k ( x 1/2 )  

2
x 2 x x
b) For what values of the constant k does f have a critical
point at x = 1? Find this value of k, determine whether f has a
relative minimum, relative maximum or neither at x = 1?
Justify your answer.
f '(1) 
k
1
 0
2 1 1
k 2
interval
f '( x)
conclusion
(0,1)
-decreasing
(1,  )
+
increasing
At x = 1, we have a relative minimum because f '( x) changes from negative to positive at x  1 .
Lesson Development: The past two lessons focused on finding the relative maximum or minimum when
given an equation. From that equation, we were able to find the derivative function and use the
derivative to find out the strictly increasing or decreasing intervals. [10 – 12 min]
The next two lessons focus on finding the relative maximum or minimum when given the graph of the
derivative function. Instead of the finding the derivative function, it’s given in the form of a graph.
How can we get the information needed to find out the strictly increasing or decreasing intervals?
Example: The graph below represents the derivative of the function f.
a) On what intervals is f increasing?
b) On what intervals is f decreasing?
f is increasing on (2, 6) and (8,10) because f '  0
f is decreasing on (0, 2) and (6,8) because f '  0
c) Find the x-coordinates of all relative minimum
of f.
d) Find the x-coordinates of all relative maximum
of f.
x  2,8 because f ' changes from – to + at these
values.
x  6 because f ' changes from = to 1 at this
value.
Note: we can’t use the graph of derivative function to determine the global maximum or minimum
without knowing the initial values. So these questions mainly ask for relative extremum.
EX1: The graph of f '( x) is given below. [6 min]
a) Identify the intervals on which f is
increasing or decreasing. Explain
(0.5, ) because f '  0 on the
interval
(, 0.5) because f '  0 on the
interval
b) Identify the critical value and
identify it as a relative maximum,
relative minimum, or neither. Explain.
There is a relative minimum at x  0.5
because f ' changes from – to + there.
EX2: The graph of f '( x) is given below. [6 min]
a) Identify the intervals on which f is increasing or decreasing. Explain
(, 0.5), (2,3) because f '  0 on the interval
(0.5, 2), (3, ) because f '  0 on the interval
b) Identify the critical value and identify it as a relative maximum, relative minimum, or neither.
Explain.
There is a relative minimum at x  2 because f ' changes from – to + there.
There is a relative maximum at x  0.5,3 because f ' changes from + to - there.
a) f is increasing on (, 2), (0, 2) and (2, )
because f '  0 .
f is decreasing on ( 2, 0) because f '  0 .
b) f attains a relative maxima at x  2 because
f ' changes from positive to negative at x  2
f attains a relative minima at x  0 because f '
changes from positive to negative at x  0
f has neither maximum nor minimum at x  2
because f ' doesn’t change sign at x  2
[5 min]
a) f attains a relative minimum at x  1 because f '( x) changes its sign (from – to +) at x  1 .
b) f attains a relative maximum at x  5 because f '( x) changes its sign (from – to +) at x  5 .
HW#34: P227 – 228: 70 – 72, 88***
***use calculator to find the critical value.
Aim: How can we use the first derivative to determine the relative extrema?
Do Now: (2007 AB-6): Let f be the function defined by f ( x)  k x  ln x for x  0 , where k is a
positive constant. [8 – 10 min]
a) Find f '( x)
b) For what values of the constant k does f have a critical point at x = 1? Find this value of k, determine
whether f has a relative minimum, relative maximum or neither at x = 1? Justify your answer.
Example: The graph below represents the derivative of the function f.
a) On what intervals is f increasing?
b) On what intervals is f decreasing?
c) Find the x-coordinates of all relative minimum
of f.
d) Find the x-coordinates of all relative maximum
of f.
EX1: The graph of f '( x) is given below. [6 min]
a) Identify the intervals on which f is
increasing or decreasing. Explain
b) Identify the critical value and
identify it as a relative maximum,
relative minimum, or neither. Explain.
EX2: The graph of f '( x) is given below. [6 min]
a) Identify the intervals on which f is increasing or decreasing. Explain
b) Identify the critical value and identify it as a relative maximum, relative minimum, or neither.
Explain.
HW#34 Solutions: P227 – 228: 70 – 72, 88
70)
71)
72)
88)