Question

The function $f\left(x\right)=x+\sin x$ has

• A. A minimum but no maximum
• B. A maximum but no minimum
• C. Neither maximum nor minimum
• D. Both maximum and minimum

Answer 1

The correct option is C

Neither maximum nor minimum

Explanation for the correct option.

$f\left(x\right)=x+\sin x$

$f’\left(x\right)=1+\cos x$

Now, $f\text{'}\left(x\right)=0$

$\Rightarrow 1+\cos x=0$

$\Rightarrow \cos x=-1$

$\Rightarrow x=\mathrm{\pi }$

Now, $f\text{'}\text{'}\left(x\right)=-\sin x$

$\Rightarrow f\text{'}\text{'}\left(\mathrm{\pi }\right)=-\sin (-\mathrm{\pi })$

$\Rightarrow f\text{'}\text{'}\left(\mathrm{\pi }\right)=0$

$\because f\text{'}\text{'}\left(\mathrm{\pi }\right)=0$, there are no maxima or minima.

Hence, option C is correct.