QJEEAdv07Mac7de: JEE Advanced 2007 Mathematics - Class 12 JEEAdv07Mac7de Solution: If a continuous function defined on the real line , 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 line $y=x$ meets $y = k{e^x}$ for $k \le 0$ at