Question

# If $$t_1=(\tan x)^{\cot x}, t_2=(\cot x)^{\cot x}, t_3=(\tan x)^{\tan x}, t_4=(\cot x)^{\tan x}, 0 < x < \dfrac{\pi}{4}$$, then:

A
t1<t2<t3<t4
B
t2>t4>t3>t1
C
t1>t4>t3>t2
D
t1>t2>t3>t4

Solution

## The correct option is D $$t_2 > t_4 > t_3 > t_1$$Since $$0 < x < \dfrac{\pi}{4}$$$$\therefore \tan x < 1$$ and $$\cot x > 1$$ .....$$(1)$$$$\therefore$$ Choose $$\tan x=1-k_1$$ and $$\cot x=1+k_2$$, where $$k_1$$ and $$k_2$$ are very small $$+$$ve quantities.$$\therefore t_3=(1-k_1)^{1-k_1}, t_1=(1-k_1)^{1+k_2}$$  ....$$(2)$$$$t_4=(1+k_2)^{1-k_1}, t_2=(1+k_2)^{1+k_2}$$  .....$$(3)$$$$\therefore t_4 > t_3$$ by $$(3)$$ and $$(2)$$, $$t_2 > t_4$$ by $$(2)$$ and $$(3)$$$$\therefore t_2 > t_4 > t_3$$. Also $$t_3 > t_1$$$$\therefore t_2 > t_4 > t_3 > t_1$$.Mathematics

Suggest Corrections

0

Similar questions
View More

People also searched for
View More