Download Find the largest open intervals where the function is concave

Find the largest open intervals where the function is
concave upward.
Answer the question.
17) The given graph is that of the derivative of a
function f. Using information obtained from
2
the graph of f' and the fact that f(-1) = - and
3
1) f(x) = x4 - 8x2
2) f(x) = 4x3 - 45x2 + 150x
f(1) =
3) f(x) = x2 + 2x + 1
2
, sketch the graph of f. Explain how
3
you obtained the graph of f.
4) f(x) = x3 - 3x2 - 4x + 5
6
y
Decide if the given value of x is a critical number for f,
and if so, decide whether the point for x on f is a relative
minimum, relative maximum, or neither.
1
5) f(x) = x2 - x - 6; x =
2
6x
-6
1
6) f(x) = (x2 - 6)(2x - 3); x =
2
-6
Find f"(x) for the function.
7) f(x) = 7x2 + 4x - 2
8) f(x) =
6
y
3x - 7
9) f(x) = 8x3 - 2x2 + 7
6x
-6
10) f(x) = 2x3/2 - 6x1/2
11) f(x) = 4x4 - 8x2 + 7
-6
12) f(x) = 5e-x2
Sketch a graph of a single function that has these
properties.
Find the indicated derivative of the function.
13) f′′′ (x) of f(x) = 6x3 + 2x2 - 2x
14) f′′′ (x) of f(x) =
18) a) Continuous and differentiable for all real
numbers
b) f′(x) > 0 on (-3 , -1) and ( 2 , ∞)
c) f′(x) < 0 on (-∞, -3) and ( -1 , 2)
d) f′′ (x) > 0 on (-∞ , -2) and ( 1 , ∞)
e) f′′ (x) < 0 on (-2 , 1)
f) f′(-3) = f′(-1) = f′(2) = 0
g) f′′ (x) = 0 at (-2 , 0) and (1, 1)
x
x+1
Find the requested value of the second derivative of the
function.
15) f(x) = 8e-x2 ; Find f′′ (2) .
16) f(x) =
ln x
;
2x
Find f′′ (1).
1
Solve the problem.
19) a) Continuous and differentiable for all real
numbers
b) f′(x) < 0 on (-∞ , -3 ) and ( 3 , ∞)
c) f′(x) > 0 on (-3 , 3)
d) f′′ (x) > 0 on (-∞ , 0 )
e) f′′ (x) < 0 on ( 0 , ∞)
f) f′(-3) = f′(3) = 0
g) An inflection point at (0,0)
26) A rectangular field is to be enclosed on four
sides with a fence. Fencing costs $7 per foot for
two opposite sides, and $8 per foot for the
other two sides. Find the dimensions of the
field of area 670 ft2 that would be the cheapest
to enclose.
27) Find the dimensions of the rectangular field of
maximum area that can be made from 500 m of
fencing material.
28) Find the dimensions that produce the
maximum floor area for a one-story house that
is rectangular in shape and has a perimeter of
155 ft.
Find the integral.
20) a) Continuous for all real numbers
b) Differentiable everywhere except x = 0
c) f′(x) < 0 on (-∞ , 0)
d) f′(x) > 0 on ( 0 , ∞)
e) f′′ (x) < 0 on (-∞ , 0) and (0, ∞)
f) f(-2) = f (2) = 5
g) y-intercept and x-intercept at (0,0)
Find the location of the indicated absolute extremum
within the specified domain.
21) Minimum of f(x) = (x2 + 4)2/3; [-2, 2]
22) Minimum of f(x) =
23) Minimum of f(x) =
24) Maximum of f(x) =
3
29)
∫7
30)
∫
31)
∫ 9x-5 dx
32)
∫ 12x3
33)
∫ 4x2/3 dx
34)
∫ (5x2 - 8x) dx
35)
∫
36)
∫
37)
∫ (3x8 - 7x3 + 5) dx
38)
∫ (9x-5 - 3x-1 ) dx
39)
∫ x2(3x + x-3) dx
1
; [-4, 1]
x+2
1 3
x - 2x2 + 3x - 4; [-2, 5]
3
x+3
; [-4, 4]
x-3
x
34
dx
x2
x dx
5
4
dx
x
x2
3 x-5
dx
x2
25) Minimum of f(x) = x3 - 3x2 ; [0, 4]
2
40)
∫ 8e4y dy
41)
∫ (t3 + e3t) dt
42)
∫
43)
∫ 9z
3z2 - 7 dz
44)
∫
x
45)
∫ (x6 - 2x5)4(6x5 - 10x4) dx
46)
∫
5x4 dx
(5 + x5 )3
47)
∫
x9
dx
ex10
48)
∫
5e1/y
dy
3y2
49)
∫ 6e4x dx
50)
∫
3e z
dz
8 z
51)
∫
ex
dx
ex + e
52)
∫ te-7t2 dt
53)
∫ 11x2e-4x3 dx
54)
∫ (1 - 6x)e3x-9x2 dx
55)
∫
56)
8 dy
(y - 9)3
(7x2 + 3)5
1
x(ln x3 )
dx
dx
3
∫
ln x7
dx
x
Answer Key
Testname: 1325-PT4
19)
1) (-∞, - 2 3/3), (2 3/3, ∞)
15
2)
,∞
4
(-∞, ∞)
(1, ∞)
Critical number, minimum at (1/2, -25/4)
Not a critical number
14
9
8) 4(3x - 7)3/2
3)
4)
5)
6)
7)
20)
9) 48x - 4
10) 1.5x-1/2 + 1.5x-3/2
11) 48x2 - 16
12) 20x2 e-x2 - 10e-x2
13) 36
14) 6(x + 1)-4
15) 112e-4
16) -
x=0
No minimum
x = -2
No maximum
x=2
27.7 ft @ $7 by 24.2 ft @ $8
125 m by 125 m
38.75 ft × 38.75 ft
21 4/3
29)
x
4
21)
22)
23)
24)
25)
26)
27)
28)
3
8
17) Graphs and explanations will vary to some degree.
The graph of f should look similar to the following.
6
y
30) 31) -
6x
-6
-6
9
4x4
+C
32)
8 9/2
x
+C
3
33)
12 5/3
x
+C
5
34)
5 3
x - 4x2 + C
3
18)
35) -
5
- 8 x+C
x
36) -
6
5
+ +C
x x
37)
1 9 7 4
x - x + 5x + C
3
4
38) 39)
4
34
+C
x
9 -4
x - 3 ln x + C
4
3 4
x + ln x + C
4
Answer Key
Testname: 1325-PT4
40) 2e4y + C
t4 e3t
41)
+
+C
4
3
42)
-4
(y - 9)2
+C
43) (3z2 - 7)3/2 + C
44)
-1
+C
56(7x2 + 3)4
45)
1 6
(x - 2x5 )5 + C
5
46) 47) -
48) -
1
2(5 + x5 )2
1
10ex10
+C
+C
5e1/y
+C
3
49)
3 4x
e +C
2
50)
3
e z+C
4
51) ln(ex + e) + C
52) -
1 -7t2
e
+C
14
53) -
11 -4x3
e
+C
12
54)
1 3x-9x2
e
+C
3
55)
1
ln ln x3 + C
3
56)
1
(ln x7)2 + C
14
5