If `x gt 0`, `y gt 0`, `z gt 0`, the least value of
`x^(log_(e)y-log_(e)z)+y^(log_(e)z-log_(e)x)+Z^(log_(e)x-log_(e)y)` is

Options

A. `3`
B. `1`
C. `5`
D. `6`

Answer

A

Explanation

`(a)` Let `log_(e)x=a`, `log_(e)y=b`, `log_(e)z=c`
`impliesx=e^(a)`, `y=e^(b)`, `z=e^(c )`
So, given expression `e^(a(b-c))+e^(b(c-a))+e^((a-b))`
Using A.M. ge G.M.
`:.(e^(a(b-c))+e^(b(c-a))+e^(c(a-b)))/(3)` ge [a^(a(b-c)+b(c-a)+c(a-b))]^(1//3)`
`:.e^(a(b-c))+e^(b(c-a)+e^(c(a-b)))ge 3`