If a continuous $f$ 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 values is negative then the equation $\mathrm{f}(\mathrm{x})=0$ has a root in R . Consider $\mathrm{f}(\mathrm{x})=\mathrm{ke}^{\mathrm{x}}-\mathrm{x}$ for all real x where k is a real constant.
The positive value of k for which $\mathrm{ke}^{\mathrm{x}}-\mathrm{x}=0$ has only one root is