Name: ________________________
Date: ___________________
The First Derivative Test For Local Extrema
(Expected Student Responses are written in red/blue)
As we have seen, any function f(x) can be graphed by generating a table of values and then
manually plotting each point individually. This process, while fairly reliable, is often long and
tedious, especially if f(x) is equal to say, x4 – 4x3 – 8x2 + 48x –2 !!! Wouldn’t it be nice if there
were some easy way to sketch this function, without going through a million and one
calculations?
Well, search no more!!! Calculus and in particular, knowledge of functions and their
derivatives, has provided us with a method of doing just that: quickly sketching complicated
functions without generating a table of values!!! As you will see, these graphs may not be
exactly “to scale” but they will possess all the essential features of the function and will allow
the reader to quickly extract any required information. Let’s get started…
1. For each of the following functions, use two DIFFERENT colors to sketch the graphs of
f(x) and f’(x) on the SAME coordinate system. In order to save yourself some time, you
may use your graphing calculator.
a) f(x) = x3 – 3x + 2
•
f’(x) = 3x2 – 3
2. Identify the turning points of the function
f(x) and classify each as either a local
maximum or a local minimum.
(-1,4)
• (1,0)
Turning points of f(x)
Local Max or Min?
(-1, 4)
Max
(1, 0)
Min
3. Divide your graph into INTERVALS by drawing a vertical line through each turning point.
Complete the following table.
x = -1
x=1
x < -1
-1< x < 1
x>1
(max)
Is the slope of the tangent to
f(x) positive(+), zero, or
negative(-) on this interval?
Is f’(x) positive (+) or
negative (-) on this interval?
(min)
+
0
-
0
+
+
0
-
0
+
Name: ________________________
Date: ___________________
4. At each of the turning points, what do you notice about the slope of the tangent to the
function f(x)?
The slope of the tangent to f(x) is equal to zero at each of the turning points.
5. At each of the turning points, what do you notice about the value of the function f’(x)?
The function f’(x) is equal to zero at each of the turning points.
6. Suggest a way of finding the turning points of a function f(x), if you are given its equation
and nothing else (without using your graphing calculator).
Set the first derivatve of the function equal to zero and solve for it roots. This
will give you the x coordinate of the turning point. Sub into the function f(x) to
find the ordered pair.
b) f(x) = x4 – 8x2 + 8
f’(x) = 4x3 – 8x
7. Identify the turning points of the function
f(x) and classify each as either a local
maximum or a local minimum.
•
(0,8)
• (-2,-8)
Turning points of f(x)
Local Max or Min?
(-2, -8)
Min
(0, 8)
Max
(2, -8)
Min
• (2,-8)
8. Divide your graph into INTERVALS by drawing a vertical line through each turning point.
Complete the following table.
Is the slope of the tangent
to f(x) positive(+), zero, or
negative(-) on this interval?
Is f’(x) positive (+) or
negative (-) on this interval?
x < -2
x = -2
(min)
-2 < x < 0
x=0
(max)
0<x<2
x=2
(min)
x>2
-
0
+
0
-
0
+
-
0
+
0
-
0
+
Examine BOTH graphs carefully before answering questions 9-12.
9. What do you notice about the values of the function f’(x) immediately before a local
max?
The function f’(x) is positive immediately before a local max.
Name: ________________________
Date: ___________________
10. What do you notice about the values of the function f’(x) immediately after a local max?
The function f’(x) is negative immediately after a local max.
11. What do you notice about the values of the function f’(x) immediately before a local min?
The function f’(x) is negative immediately before a local min.
12. What do you notice about the values of the function f’(x) immediately after a local min?
The function f’(x) is positive immediately after a local min.
13. Using your own words, write a rule that will allow you to determine whether a turning
point is a local maximum or a local minimum based on the values of the derivative f’(x).
For any turning point (c, f(c)):
If the function f’(x) changes sign from positive to negative then c is a local maximum.
If the function f’(x) changes sign from negative to positive then c is a local minimum.
14. Your new rule needs a name. What are you going to call your new rule?
The crazy extreme rule (get it? loca extrema rule? CHEEEESY!!!)
15. Repeat for the following function.
c) f(x) = 2x3 + 3x2 – 12x
f’(x) = 6x2 + 6x – 12
16. Identify the turning points of the function
f(x) and classify each as either a local
maximum or a local minimum.
(-2,20)
•
Turning points of f(x)
Local Max or Min?
(-2, 20)
Max
(1, -7)
Min
• (1,-7)
17. Divide your graph into INTERVALS by drawing a vertical line through each turning point.
Complete the following table.
x < -2
x = -2
-2 < x < 1
x=1
x>1
Is the slope of the tangent to
f(x) positive(+), zero, or
negative(-) on this interval?
Is f’(x) positive (+) or
negative (-) on this interval?
+
0
-
0
+
+
0
-
0
+
Name: ________________________
Date: ___________________
18. Does your rule apply to this function? If not, return to question 13 and modify your rule.
Yes the rule works, so no need to modify. Yipee!!!
19. Now that you know how to find and classify turning points, you can work backwards
(starting with the turning points) to graph the function. Try it! Given the following
information and without using your graphing calculator, sketch the graph of f(x).
f’(x) = 12x2 – 27
d) f(x) = 4x3 - 27x + 10
x < -1.5
x = -1.5
-1.5 < x < 1.5
x = 1.5
x > 1.5
Is the slope of the tangent
to f(x) positive(+), zero, or
negative(-) on this interval?
Is f’(x) positive (+) or
negative (-) on this interval?
20. In the table above, where did the values x = -1.5 and x = 1.5 come from? How would you
get these values?
These values are the turning points of the function f(x) which also happen
to be the roots of the function f’(x). They can be found by setting the
derivative equal to zero and solving for x.
21. Start by identify the turning points of the
function f(x) and classify each as either
a local maximum or a local minimum.
Turning points of f(x)
Local Max or Min?
(-1.5, 37)
Max
(1.5, -17)
Min
(-1.5,37)
•
• (1.5,-17)
22. Use your graphing calculator to check your answer.
23. Were you able to correctly apply your rule to come up with the graph of f(x)? If yes, try to
explain why your rule works. If not, then try to explain why your rule does not work.
Yes the rule works, so no need to modify. Yipee!!!
Name: ________________________
Date: ___________________
24. Given a function f(x), write down the steps that you would follow in order to quickly
sketch the graph of f(x).
1. find f’(x)
2. find the turning points by setting f’(x) = 0 and solving for roots
3. plot the turning points
4. divide the graph into intervals by drawing a line through each turning
point
5. Create a table and check to see if f’(x) is positive or negative on each
interval
6. Using the information from step 4 determine whether the critical point is a
local max or a local min.
7. Find and plot the intercepts
8. draw a smooth curve through all intercepts and turning points such that
local max/min conditions are satisfied.
25. Write a short paragraph summarizing the information contained in this investigation.
Please not that you may be required to share your summary with the class.
Answers will vary. Students will write their responses on the board in a
roundtable-type format. Check to see that nothing has been omitted.
26. Try graphing the following functions using your new rule and the steps outlined in
question 23. Only use your graphing calculator to check your answers. Does your rule
work for all functions? If not, return to question 13 and modify your rule.
i)
ii)
iii)
f(x) = -2x5 + 5
f(x) = 3x4 – 4x3
f(x) = 2x4 - 8x3 + 9x2
iv)
v)
f(x) = x3 (whoops)
f(x) = x3 – 12x + 4
Modification to rule: Solutions of f’(x) do not necessarily yield turning points. All
max/min points are necessarily turning points but not all turning
points are necessarily max/min points.