In the interval `(0,pi//2)` the fucntion `f(x)= tan^nx+cot^n`x attains

Options

A. the minimum value which is independent of n
B. a minimum vlaue which is a fuction of n
C. the minimum vlaue which is a function of 1
D. non of these

Answer

A

Explanation

we have
`f(x)=tan^n x+ cot^n x`
`rArr f(x)= (tan^(n//2)x-cot^(n//2)x)^2+ 2 ge 2`
Cleary f(x) attains the minimum value 2 when `tan^(n//2)x-cot ^(n//2)x=0 i.e x = pi/4`