Q. 123.9( 35 Votes )
Which of the following functions are strictly decreasing on 
?
A. cos x
B. cos 2x
C. cos 3x
D. tan x
Answer :
(A) Let f1(x) = cosx
In interval
, 
Therefore, f1(x) = cosx is strictly decreasing in interval
.
(B) Let f2(x) = cos2x
Now, 0 < x < 
⇒ 0 < 2x < π
⇒ sin2x > 0
⇒ -2sin2x < 0
Therefore, f2(x) = cos2x is strictly decreasing in interval
.
(C) Let f3(x) = cos3x
Now, 
⇒ sin3x = 0
⇒ 3x = π, as xϵ
⇒ x = 
The point x =
divides the interval 
into two distinct intervals.
i.e. 
and 
Now, in interval, 
,
f3'(x) = -3sin3x < 0 as (0 < x < 
=> 0 < 3x < π)
Therefore, f3 is strictly decreasing in interval
.
Now, in interval 
f3'(x)=-3sin3x > 0 as 
Therefore, f3 is strictly increasing in interval
.
(D) Let f4 = tanx
In interval
,
Therefore, f4 is strictly increasing in interval 
.
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