Let 
be a twice differentiable function such that 
for all 
and 
, where 
is a real number. Let 
.
Consider the following two statements:
(I) 
is increasing in 
(II) 
is decreasing in 
. Then,
Text Solution
Verified by ExpertsA
Given 
for all 
, which means 
is a strictly increasing function.
We are given 
. Since 
is strictly increasing, 
for 
and 
for 
.
The function 
is defined as 
for 
.
Let 
.
Differentiating 
with respect to 
:
.
Case 1: 
.
In this interval, 
, so 
.
Also, 
, so 
.
Since 
is increasing and 
, for 
, we have 
.
Thus, 
. So 
is decreasing in 
.
Statement (I) is False.
Case 2: 
.
In this interval, 
, so 
.
Also, 
, so 
.
Thus, 
.
Then 
. So 
is increasing in 
.
Statement (II) is False.
Therefore, neither (I) nor (II) is true.
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