1. Tardigrade
  2. Question
  3. Mathematics
  4. 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

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Q. 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

Application of Derivatives

Solution:

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