If a continuous function `f` defined on the real line R assume positive and negative values in R, then the equation `f(x)=0` has a root in R. For example, if it is known that a continuous function `f` on R is positive at some point and its minimum value is negative, then the equation `f(x)=0` has a root in R. Consider `f(x)= ke^(x)-x`, for all real x where k is a real constant.
For k > 0, the set of all values of k for which `y=ke^(x)-x=0`has two distinct roots is

Options

A. `(0,1//e)`
B. `(1//e,1)`
C. `(1//e,oo)`
D. (0,1)

Answer

A

Explanation

Let f(x)`=ke^(x)-x,k gt 0`.
f(x)=0 will have two distinct roots, if its minimum value is negative.
i.e. `1+log k lt 0 rArr lt (1)/(e ) rArr k in (0,1//e) " "[:' k gt 0]`