Question

Consider the following statements :
1. The function $f\left(x\right)=x^2+2\cos x$ is increasing in the interval $(0,\pi )$
2. The function $f\left(x\right)=ln(\sqrt{1+x^2}-x)$ is decreasing in the interval $(-\infty ,\infty )$
Which of the above statements is\are correct?

• A. 1 only
• B. 2 only
• C. Both 1 and 2
• D. Neither 1 nor 2

Answer 1

The correct option is C Both 1 and 2

Solution:

1. We have, $f\left(x\right)=x^2+2\cos x$

On differentiating, we get $f^{\prime }\left(x\right)=2x-2\sin x$

Here, $f^{\prime }\left(x\right)>0$ in the interval $(0,\pi )$

$\therefore f\left(x\right)$ is increasing in the interval $(0,\pi ).$

2. We have, $f\left(x\right)=ln(\sqrt{1+x^2}-x)$

On differentiating, we get

$f^{\prime }\left(x\right)=\frac{1}{\sqrt{1+x^2}-x}\times (\frac{x}{\sqrt{1+x^2}}-1)$

Here, $f^{\prime }\left(x\right)<0$ in the interval $(-\infty ,\infty )$

$\therefore f\left(x\right)$ is decreasing in the interval $(-\infty ,\infty ).$

Hence, C is the correct option.