The minimum value of the function `f(x) =tan(x +pi/6)/tanx` is:

Options

A. 1
B. 0
C. `(1)/(2)`
D. 3

Answer

D

Explanation

`f(x)=(tan(x+(pi)/(6)))/(tanx)=cot xtan(x+(pi)/(6))`
`f'(x)=cot x sec^(2)(x+(pi)/(6))-"cosec"^(2)x tan(x+(pi)/(6))`
`therefore" "f''(x)=2 cot x sec^(2)(x+(pi)/(6))tan(x+(pi)/(6))`
`-"cosec"^(2)xsec^(2)(x+(pi)/(6))`
`="cosec"^(2)x sec^(2)(x+(pi)/(6))`
`+2 "cosec"^(2)x cot x tan(x+(pi)/(6))`
`"Now "f'(x)=0`
`rArr" "(1)/(2)sin 2x=(1)/(2)sin(2x+(pi)/(3))`
`rArr" "2x=pi-2x-(pi)/(3)`
`rArr" "x=(pi)/(6)`
Thus `f''(pi//6)gt0`
`therefore " At "x=(pi)/(6),f(x)` is minimum and there is no other minimum in `(0,(pi)/(2))`.
`therefore" The minimum vlaue of f(x)"=f((pi)/(6))=(tan((pi)/(6)+(pi)/(6)))/(tan.(pi)/(6))=(sqrt3)/((1)/(sqrt3))=3`