Question

It is a continuous function $f$ defined on the real line $R$, assume positive and negative values in $R$ then the equation $f\left(x\right)=0$ has 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$f\left(x\right)=0$ has a root in $R$. Consider$f\left(x\right)=k{e}^x–x$ for all real $x$ where$k$ is a real constant.The line $y=x$ meets $y=k{e}^x$ for $k\le 0$ at

• A. No point
• B. One point
• C. Two points
• D. More than two points

Answer 1

The correct option is B

One point

Explanation for the correct answer:

Finding the point where the line meets:

$f\left(x\right)=k{e}^x–x$

$f\text{'}\left(x\right)=k{e}^x–1<0$ if $k\le 0$ [Given]

$f\left(x\right)$ is decreasing for all $x$.

Now$li{m}_{x\to \infty }k{e}^x–x=–\infty$

Also$f\left(k\right)=k(e^k–1)\ge 0[\text{because}k\le 0\Rightarrow e^k–1\le 0]$

$f\left(x\right)=0$, has exactly one root in $[k,\infty ]$

$y=x$ meets $y=k{e}^x$ for $k\le 0$ at one point.

Therefore, the correct answer is option (B).