QJEEAdv07Ma7b90: JEE Advanced 2007 Mathematics - Class 12 JEEAdv07Ma7b90 Solution: If a continuous function f defined on the real line R , assumes …

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 $$f\left(x\right)=k{e}^x-x$$ for all real $$x$$ where $$k$$ is real constant.

The positive value of $$k$$ for which $$k{e}^x-x=0$$ has only one root is