Let $\theta \in\left(0, \frac{\pi}{4}\right)$ and $t_{1}=(\tan\theta)^{\tan\theta}, t_{2}=(\tan\theta)^{\cot\theta}, t_{3}=(\cot\theta)^{\tan\theta}$ and $t_{4}=(\cot\theta)^{\cot\theta}$, then

(B) $t_{4}>t_{3}>t_{1}>t_{2}$

(A) $t_{1}>t_{2}>t_{3}>t_{4}$

(C) $t_{3}>t_{1}>t_{2}>t_{4}$

(D) $t_{2}>t_{3}>t_{1}>t_{4}$

JEE Advanced · 2006 · 09 Apr 2006 · Application of Derivatives

Let

θ \in (0, \frac{π}{4})

and

t_{1} = (tan θ)^{t a n θ}, t_{2} = (tan θ)^{c o t θ}, t_{3} = (cot θ)^{t a n θ}

and

t_{4} = (cot θ)^{c o t θ}

, then

Answer: (B)

For

θ \in (0, π /4)

, we have

0 < tan θ < 1

and

cot θ > 1

.

Let

a = tan θ < 1

and

b = cot θ > 1

.

Then:

Since

0 < a < 1

a^{b} < a^{a} < a^{0} = 1

(smaller exponent gives larger value).

Since

b > 1

b^{0} = 1 < b^{a} < b^{b}

(larger exponent gives larger value).

Comparing:

b^{b} > b^{a} > 1 > a^{a} > a^{b}

.

Therefore,

t_{4} > t_{3} > t_{1} > t_{2}

JEE Advanced · 2006 · 09 Apr 2006 · Single correct option