Let $$\mathrm{A_{1}, G_{1}}, \mathrm{H}_{1}$$ denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For $$n \geq 2$$, let $$\mathrm{A}_{n-1}$$ and $$\mathrm{H}_{n-1}$$ have arithmetic, geometric and harmonic means as $$\mathrm{A_{n}}$$, $$\mathrm{G}_{\mathrm{n}}, \mathrm{H}_{\mathrm{n}}$$ respectively.

Which one of the following statements is correct?

If a continuous functions $$f$$ 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 $$f(x)=k e^{x}-x$$ for all real $$x$$ where $$k$$ is a real constant.

The line $$y=x$$ meets $$y=k e^{\mathrm{x}}$$ for $$k \leq 0$$ at

If a continuous functions $$f$$ 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 $$f(x)=k e^{x}-x$$ for all real $$x$$ where $$k$$ is a real constant.

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

If a continuous functions $$f$$ 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 $$f(x)=k e^{x}-x$$ for all real $$x$$ where $$k$$ is a real constant.

For $$k > 0$$, the set of all values of $$k$$ for which $$k e^{x}-x=0$$ has two distinct roots is