Calculus AB
Section ________
~Wks 3-6B Do not use your calculator!
Name __________________
Date ___________________
1. Given f  x   x3  5 x 2  8 x  7. Use the Second Derivative Test to find whether f has a local
maximum or a local minimum at x =  4. Justify your answer.
2. Given f  x   3 x  2sin x. Use the Second Derivative Test to find whether f has a local
maximum or a local minimum at x =

. Justify your answer
6
On problems 3 – 5, find the critical points of each function, and determine whether they are relative
maximums or relative minimums by using the Second Derivative Test whenever possible.
3. f  x   x3  3x 2  3
4
x
5. f  x   sin x  cos x, 0  x  2
________________________________________________________________________________
6. Consider the curve given by x 2  4 y 2  7  3xy.
dy 3 y  2 x
(a) Show that

.
dx 8 y  3x
(b) Show that there is a point P with x-coordinate 3 at which the line tangent to the curve at P
is horizontal. Find the y-coordinate of P.
d2y
(c) Find the value of
at the point P found in part (b). Does the curve have a local
dx 2
maximum, a local minimum, or neither at point P? Justify your answer.
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7. For what values of a and b does the function f  x   x3  ax 2  bx  2 have a local
4. f  x   x 
maximum when x   3 and a local minimum when x  1 ?
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8. The graph of a function f is shown on the right.
Fill in the chart with +,  , or 0.
y
Point
A
B
C
f
f
f 
B
A
x
C
9. The function h is defined by h  x   f  g  x   , where f and g are the functions whose
graphs are shown below.
(a) Evaluate h  2  .
(b) Estimate h 1 .
(c) Is the graph of the composite function h increasing or decreasing at x = 3? Show your
reasoning.
(d) Find all values of x for which the graph of h has a horizontal tangent. Show your reasoning.
Selected Answers to Worksheet 3-6B on Derivatives
1. f has a local maximum at x   4 since f    4  0 and f    4  14. Concave down
2. f has a local minimum at x 
3. Crit. pts:  0, 3 and  2, 1

 
 
since f     0 and f     1. Concave up
6
6
6
 0, 3 is a relative maximum because f   0  0 and f   0   6. Concave down
 2, 1 is a relative minimum because f   2  0 and f   2  6. Concave up
4. Crit. pts:   2,  4 and  2, 4
  2,  4 is a relative maximum because f    2  0 and f    2  1. Concave down
(2, 4) is a relative minimum because f   2  0 and f   2  1. Concave up
 3

 7

, 2
5. Crit. pts:  ,  2  and 
 4

 4

 3

 3 
 3 
 , 2  is a relative maximum because f     0 and f      2. Concave down
 4 
 4 
 4

 7

 7 
 7 
, 2  is a relative minimum because f  

  0 and f  
  2. Concave up
 4 
 4 
 4

7. a = 6, b = 9
8.
Point
A
B
C
f
+

f
+
f 

+