If `x_1a n dx_2` are the roots of the equation `e^2 x^(lnx)=x^3` with `x_1> x_2,` then `x_1=2x_2` (b) `x_1=x2 2` `2x_1=x2 2` (d) `x1 2=x2 3`

Options

A. `x_(1) = 2x_(2)`
B. `x_(1) = x_(2)^(2)`
C. `2x_(1) = x_(2)^(2)`
D. `x_(1)^(2) = x_(2)^(3)`

Answer

B

Explanation

`e^(2)*x^("In "x) = x^(3)`
Taking log on both sides, we get
In ` (e^(2)*x^("In "x)) ="In "(x^(3))`
`rArr ("In "x)^(2) - 3" In " x + 2 = 0`
` rArr ("In " x - 2)("In " x -1) = 0`
If In ` x = 2 rArr x = e^(2)`
If In ` x = 1 rArr x = e`
Since` x_(1) gt x_(2)," we get " x_(1) = e^(2) and x_(2) = e`
` rArr x_(2)^(2) = x_(1)`