194
CHAPTER 23
Fig. 23-8
Setting /=0, we find 3x
2 - Wx + 3 = 0, (3*- l)(;t-3) = 0, x=| or x = 3. By the secondderivative test, we have a relative minimum at jt = 3, and a relative maximum at x=\. As x—>±°°,
Note that In
In
In
and
23.77
Prove that In x < Vx for x > 0.
By the second-derivative test, the function v = Vx — In x has an absolute minimum value of 2 - 2 In 2
when x=4. Therefore, Vx - Inx>2 -21n2>0 for all *>0. Hence, Vx>lnx for x>0.
23.78
Evaluate
The latter is an approximating sum for
In 3 - In 2.
Hence, the limit is
In
23.79
Find
Divide numerator by denominator, obtaining
Hence, the integral reduces to
23.80
Find
Let
u
2 = x + ll, 2udu = dx. Then the integral becomes
In
In
In
>
CHAPTER 23
Fig. 23-8
Setting /=0, we find 3x
2 - Wx + 3 = 0, (3*- l)(;t-3) = 0, x=| or x = 3. By the secondderivative test, we have a relative minimum at jt = 3, and a relative maximum at x=\. As x—>±°°,
Note that In
In
In
and
23.77
Prove that In x < Vx for x > 0.
By the second-derivative test, the function v = Vx — In x has an absolute minimum value of 2 - 2 In 2
when x=4. Therefore, Vx - Inx>2 -21n2>0 for all *>0. Hence, Vx>lnx for x>0.
23.78
Evaluate
The latter is an approximating sum for
In 3 - In 2.
Hence, the limit is
In
23.79
Find
Divide numerator by denominator, obtaining
Hence, the integral reduces to
23.80
Find
Let
u
2 = x + ll, 2udu = dx. Then the integral becomes
In
In
In
>