In `!ABC` , least value of `(e^(A))/A+(e^(B))/B+(e^(C))/C` is equal to

Options

A. `9/pie^(pi//3)`
B. `pi/3e^(pi//3)`
C. `pi/9e^(pi//3)`
D. none of these

Explanation

We have,
`(e^(A))/A+(E^(B))/B+(e^(C))/Cge3((e^(A+B+C))/(ABC))^(1//3)` [Using AM `ge` GM]
`rArr`(e^(A))/A+(E^(B))/B+(e^(C))/Cge3((e^(pi))/(ABC))^(1//3)`
Also,
`A+B+Cge3(ABC)^(1//3)` [Using AM gt GM]
`rArrpi/3ge(ABC)^(1//3)`
`rArr(1/(ABC))^(1//3)ge3/(pi)`
`rArr((e^(pi))/(ABC))^(1//3)ge3/(pi)e^(pi//3)`
`rArr3((e^(pi))/(ABC))^(1//3)ge9/pie^(pi//3)rArr(e^(A))/A+(e^(B))/B+(e^(C ))/Cge9/pie^(pi//3)`