Question

Let $f:[1,\infty )\to R$ be a function defined by $f\left(x\right)=x\underset{1}{\overset{x}{\int }}\frac{e^t}{t}dt-e^x$

I. $f\left(x\right)$ is an increasing function.
II. $\underset{x\to \infty }{lim}f\left(x\right)=0$

Which of the following statements are correct?

• A. Only I
• B. Only II
• C. Both I and II
• D. Neither of them is correct

Answer 1

The correct option is A Only I

$f\left(x\right)=x\underset{1}{\overset{x}{\int }}\frac{e^t}{t}dt-e^x\Rightarrow f^{\prime }\left(x\right)=\underset{1}{\overset{x}{\int }}\frac{e^t}{t}dt+x\times \frac{e^x}{x}-e^x$
$\Rightarrow f^{\prime }\left(x\right)=\underset{1}{\overset{x}{\int }}\frac{e^t}{t}dt\ge 0\forall x\in [1,\infty )$
This means $f\left(x\right)$ is an increasing function.
So, $\underset{x\to \infty }{lim}f\left(x\right)\to \infty$