If a continuous function f defined on the real line R, assumes 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 ke x - x =0 has two distinct roots is

Question

If a continuous function f defined on the real line R, assumes 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 ke x - x =0 has two distinct roots is (A) (0,1 / e ) (B) (1 / e , 1) (C) (1 / e , ∞) (D) (0,1)

Tardigrade Admin · Accepted Answer

k >0 for two distinct roots k ∈(0, (1/ e ))

$(0, 1/ e)$

$(1/ e, 1)$

$(1/ e, \infty)$

$(0, 1)$

$k > 0$ for two distinct roots $k \in (0, \frac{1}{e})$

(0,1/e)

(1/e,1)

(1/e,∞)

(0,1)

k>0 for two distinct roots k∈(0,e1​)

$(0, 1/ e)$

$(1/ e, 1)$

$(1/ e, \infty)$

$(0, 1)$

$k > 0$ for two distinct roots $k \in (0, \frac{1}{e})$