Which of the following functions are strictly decreasing on (0,π/2)?
Which of the following functions are strictly decreasing on (0,π/2)?
- cos x
- cos 2x
- cos 3x
- tan x
The Correct Option is A, B
Solution and Explanation
(A) Let f1(x) = cos x.
=f1(x) = -sin x
In interval (0,\(\frac \pi2\)), f'1(x) = -sin x<0.
\(\implies\)f1(x) = cos x is strictly decreasing in interval (0,\(\frac \pi2\)).
(B) Let f2(x) = cos2x.
f'2(x) = -2sin 2x
Now 0<x<\(\frac \pi2\) \(\implies\) 0<2x<π \(\implies\) sin 2x>0 \(\implies\) -2 sin 2x<0
f'1(x) = -2sin 2x < 0 on (0,\(\frac \pi2\))
\(\implies\)f'2(x) = cos 2x is strictly decreasing in interval (0,\(\frac \pi2\)).
(C) Let f3(x) = cos3x.
f'3(x) = -sin3x
Now, f'3(x) = 0
\(\implies\)sin3x=0\(\implies\)3x=π, as x ε (0,\(\frac \pi2\))
\(\implies\)x = \(\frac \pi2\)
The point x=\(\frac \pi3\) divides the interval (0,\(\frac \pi2\)) into two disjoint intervals
i.e., 0 (0,\(\frac \pi3\)) and (\(\frac \pi3\),\(\frac \pi2\)).
Now, in interval(0,\(\frac \pi3\)), f3(x) =-3 sin3x<0 [as 0<x<\(\frac \pi3\)=0<3x<\(\pi\)].
∴ f3 is strictly decreasing in interval (0,\(\frac \pi3\))
However, in interval (\(\frac \pi3\),\(\frac \pi2\)), f3(x)=-3sin 3x>0 [as \(\frac \pi3\)<x<\(\frac \pi2\)\(\implies\)π<3x<\(\frac {3\pi}{2}\)]
∴ f3 is strictly increasing in interval (\(\frac \pi3\),\(\frac \pi2\)).
Hence, f3 is neither increasing nor decreasing in interval (0,\(\frac \pi2\))
(D) Let f4(x) = tan x.
\(\implies\)f4(x) = sec2 x
In interval (0,\(\frac \pi2\)), f'4(x) = sec2x>0.
f4 is strictly increasing in interval (0,\(\frac \pi2\)).
Therefore, functions cos x and cos 2x are strictly decreasing in (0,\(\frac \pi2\)).
Hence, the correct answers are (A) and (B).
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Concepts Used:
Increasing and Decreasing Functions
Increasing Function:
On an interval I, a function f(x) is said to be increasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) ≤ f(y)
Decreasing Function:
On an interval I, a function f(x) is said to be decreasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) ≥ f(y)
Strictly Increasing Function:
On an interval I, a function f(x) is said to be strictly increasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) < f(y)
Strictly Decreasing Function:
On an interval I, a function f(x) is said to be strictly decreasing, if for any two numbers x and y in I such that x < y,
⇒ f(x) > f(y)
Graphical Representation of Increasing and Decreasing Functions
