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Concavity and the Second Derivative Test Worksheet

Concavity and the Second Derivative Test Worksheet
  • Page 1
 1.  
Find the interval in which f (x) = xe6x is concave upward.
a.
(1 2, ∞)
b.
(- ∞, - ∞)
c.
(- 1 3, ∞)
d.
(- ∞, -1 2)


Solution:

f(x) = xe6x

f ′(x) = 6x e6x + e6x
[First derivative.]

f ¢¢(x) = 36x e6x+ 6e6x + 6e6x = e6x(36x + 12)
[Second derivative.]

f(x) is concave upwards if f ¢¢(x) > 0

e6x (36x + 12) > 0

36x + 12 > 0
(e6x > 0.)

x > - 1 / 3 x (- 1 / 3, ∞)
[Solve the inequality.]


Correct answer : (3)
 2.  
Find the interval in which f(x) = 7x(ln 8x) is concave upward.
a.
(0, ∞)
b.
(- 7 , 7 )
c.
(- ∞, ∞)
d.
(- ∞, 0)


Solution:

f(x) = 7x(ln 8x)

f ′(x) = 7 + 7(ln 8x)
[First derivative.]

f ¢¢(x) = 7x
[Second derivative.]

f (x) is concave upward if f ¢¢(x) > 0

Þ 7x > 0

x > 0 x (0, ∞)
[Solve the inequality.]


Correct answer : (1)
 3.  
What is the interval in which f(x) = (ln(8x)) (x) is concave upward ?
a.
(- ∞, ∞)
b.
(- ∞, 1 8e32)
c.
(- 1 8e32, ∞)
d.
(1 8e32, ∞)


Solution:

f(x) = (ln(8x)) / (x)

Domain of f(x) is (0, ∞)
[Evaluate.]

f ′(x) = (1 - ln 8x) /x2
[First derivative.]

f ¢¢(x) = [x2(-88x) - (1 - ln 8x)(2x)]x4

f ¢¢(x) =(- x - 2x + 2x ln 8x)x4

f ¢¢(x) = [x(2ln 8x - 3)]x4
[Second derivative.]

f(x) is concave upward if f ¢¢(x) > 0

[x(2ln 8x - 3)]x4 > 0

2ln 8x - 3 > 0
[x > 0.]

ln 8x > 3 / 2

x > 1 / 8e32 x (1 / 8e32, ∞)
[Solve the inequality.]


Correct answer : (4)
 4.  
What is the interval in which f (x) = 3x3 + 27x2 + 12x + 40 is concave downward?
a.
(- ∞, - 3)
b.
(- 3, ∞)
c.
(- ∞, 3)
d.
(- ∞, ∞)


Solution:

f(x) = 3x3 + 27x2 + 12x + 40
[Given.]

f ′(x) = 9x2 + 54x + 12
[First derivative.]

f ¢¢(x) = 18x + 54 = 18(x + 3)
[Second derivative.]

f(x) is concave downward if f ¢¢(x) < 0

18(x + 3) < 0

x < - 3 x (- ∞, - 3)
[Solve the inequality.]


Correct answer : (1)
 5.  
Find the interval in which f (x) = 5x - 41x is concave downward.
a.
(- ∞, 0)
b.
(- ∞, 82)
c.
(- ∞, ∞)
d.
(0, ∞)


Solution:

f(x) = 5x - 41x
[Given function.]

f ′(x) = 5 + 41x2
[First derivative.]

f ¢¢(x) = - 82x3
[Second derivative.]

f(x) is concave downward if f ¢¢(x) < 0

- 82x3 < 0

82x3 > 0

x3 > 0 x > 0 x (0, ∞)
[Solve the inequality.]


Correct answer : (4)
 6.  
Find the interval in which f (x) = e- 3x is concave downward.
a.
(3, ∞)
b.
(- ∞, ∞)
c.
Not existing
d.
(- ∞, 9)


Solution:

f (x) = e- 3x
[Given.]

f ′(x) = - 3e- 3 x
[First derivative.]

f ¢¢(x) = 9e- 3x
[Second derivative.]

Since for all real values of x, 9e- 3x > 0, f (x) is concave upward for all x (- ∞, ∞).

So there is no real value of x for which f (x) is concave downward.


Correct answer : (3)
 7.  
In (0, π2), f (x) = 3x cos x + 36 is
a.
Concave downward
b.
Concave upward in(0, π4) & concave downward in (π4, π2)
c.
Concave downward in(0, π4) & concave upward in (π4, π2).
d.
Concave upward


Solution:

f(x) = 3xcosx + 36

f ′(x) = 3cos x - 3x sin x
[First derivative.]

f ¢¢(x) = - 3sin x - 3sin x - 3x cos x = - 3(x cos x + 2sin x)
[Second derivative.]

For all x (0, π2), 3(x cosx + 2sinx) > 0

- 3(x cos x + 2sin x) < 0 f ¢¢(x) < 0

So, f (x) is concave downward in (0, π2)


Correct answer : (1)
 8.  
Find the largest interval in which f (x) = 18x4 + 6x2 + 44 is concave upward.
a.
(0, ∞)
b.
(- ∞, ∞)
c.
(- ∞, 12)
d.
(18, ∞)


Solution:

f(x) = 18x4 + 6x2 + 44
[Given function.]

f ′(x) = 72x3 + 12x
[First derivative.]

f ¢¢(x) = 216x2 + 12 = 12(18x2 + 1)
[Second derivative.]

For all the real values of x, x2 > 0 12(18x2 + 1) > 0 f ¢¢ (x) > 0

So, f(x) is concave upward for all x (- ∞, ∞)


Correct answer : (2)
 9.  
For the function f (x) = 8ax2 + 2bx + 47 where a ≠ 0, which of the following is true?
a.
If a < 0, f (x) is concave upward
b.
If b > 0, f (x) is concave downward
c.
If b < 0, f (x) is concave upward
d.
If a > 0, f (x) is concave upward


Solution:

f(x) = 8ax2 + 2bx + 47 where a ≠ 0

f ′(x) = 16ax + 2b
[First derivative.]

f ¢¢(x) = 16a
[Second derivative.]

If a > 0, f ¢¢ (16x) > 0 hence f (x) is concave upward for a > 0.


Correct answer : (4)
 10.  
What is the interval in which f(x) = x - 8x - 7 is concave downward ?
a.
(0, ∞)
b.
(- ∞, 7 )
c.
(- ∞, ∞)
d.
(7, ∞)


Solution:

f (x) = x - 8x - 7

f ′(x) = [(x - 7)(1) - (x - 8)(1)](x - 7)2 = 1(x - 7)2
[First derivative.]

f ¢¢(x) = - 2(x - 7)3
[Second derivative.]

f(x) is concave downward if f ¢¢(x) < 0

- 2(x - 7)3 < 0

2(x - 7)3 > 0

(x - 7) 3 > 0

x - 7 > 0 x > 7 x (7, ∞)
[Solve the inequality.]


Correct answer : (4)

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