Paragraph: If a continuous $f$ defined on the real line $R$, assume 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 $f(x)=0$ has a root in $R$. Consider $f(x)=k e^x-x$ for all real $x$, where $k$ is real constant.Question: The line $y=x$ meets $y=k e^x$ for $k \leq 0$ at

• A

  no point
• Correct answer
  B

  one point
• C

  two points
• D

  more than two points

Solution

$$ \text { Let } y=x \text { intersect the curve } y=k e^x \text { at exactly one point when } k \leq 0 \text {. } $$

Hence, the correct answer is option: 'B'.