AP Calculus
Name: _________________________
WS 5.1 Absolute Extrema
1. Explain the difference between an absolute minimum and a local minimum.
_____________________________________________________________________________________________
3. Use the graph to state the absolute and local
2. For each of the numbers a, b, c, d , e, r , s, t ,
maximum and minimum values of the function.
state whether the function whose graph is shown
has an absolute maximum or minimum, a local
maximum or minimum, or neither a maximum or
minimum.
_____________________________________________________________________________________________
Find the critical numbers.
4.
f ( x)  5  6 x  2 x 3
6. f ( x)  5 x
2
3
x
5
3
5. f ( x) 
7.
x
x 4
2
f ( x)  sin 2 (2 x),0  x  2
_____________________________________________________________________________________________
8. The graph of a function defined on the interval 0  x  5 is shown. Which of the following statements is true
on the interval 0  x  5 ?
(A)
(B)
(C)
(D)
Both the Intermediate Value and Extreme Value Theorems apply
Only the Intermediate Value Theorem applies
Only the Extreme Value Theorem applies
Neither the Intermediate Value nor the Extreme Value Theorems apply
_____________________________________________________________________________________________
9. Which of the following graphs of functions satisfy the hypotheses of both the Extreme Value Theorem and the
Intermediate Value Theorem on the closed interval [ a, b] ?
(A)
(B)
(C)
(D)
graph 1 only
graphs 2 and 5
graphs 3 and 5
graphs 1, 2, and 4
Find the absolute maximum and absolute minimum values of
10.
f ( x)  x3  3x  1,[0,3]
11.
f on the given interval.
f ( x)  4 x3  x 2  4 x  1,[1,2]
12.
 
f ( x)  sin x  cos x, 0, 
 3
_____________________________________________________________________________________________
13. What is the maximum acceleration attained on the interval 0  t  3 by the particle whose velocity is given
by v(t )  t  3t  12t  4 ?
3
2
_____________________________________________________________________________________________
14. Let g be the function given by
point at x 
g ( x)  x 2ekx
, where k is a constant. For what value of k does g have a critical
2
?
3
(A) -3
(B) 
3
2
(C) 
1
3
(D) 0
(E) there is
no such k
_____________________________________________________________________________________________
15. Given the function
(A) 0 only
f ( x)  x 2e2 x . For what value(s) of x does f have a critical point?
(B) -1 only
2
(C) e
(D) -1 and 0
(E) None of these
_____________________________________________________________________________________________
16. Given the function
(A) 0 only
f ( x)  e x ( x 2  3) . For what value(s) of x does f have a critical point?
(B) 3 only
(C)  3
(D) -3 and 1
(E) None of these
_____________________________________________________________________________________________
17. Given the function f ( x)  sin x  2 . What is the absolute maximum value of f on the interval
0  x  2 ?

2
(A) 0
(B)
(C) 1
(D) 3
(E) 2