Thank you for reporting, we will resolve it shortly
Q.
On which of the following intervals is the function $x^{100}+sin \, x-1$ decreasing?
NTA AbhyasNTA Abhyas 2020Application of Derivatives
Solution:
$f \left(x\right) = x^{100} + sin x - 1$
$\Rightarrow f^{'}\left(x\right)=100x^{99}+cosx$
If $0 < x < \frac{\pi }{2} ,$ then $f^{'} \left(x\right) > 0 ,$ , therefore $f \left(x\right)$ is increasing on $\left(0 , \frac{\pi }{2}\right) .$
If $0 < x < 1 ,$ then
$100x^{99}>0$ and $cosx>0$ [ $\because x$ lies between $0$ and $1$ ]
$\Rightarrow f^{'}\left(x\right)=100x^{99}+cosx>0$
$\Rightarrow f\left(x\right)$ is increasing on $\left(0 , 1\right).$
If $\frac{\pi }{2} < x < \pi ,$ then
$100 x^{99} > 100 \, \left[\because x > 1 \Rightarrow x^{99} > 1\right]$
$\Rightarrow 100 x^{99} + cos x > 0 \, \left[\because cos x \geq - 1 \Rightarrow 100 x^{99} + cos x > 99\right]$
$\Rightarrow f^{'} \left(x\right) > 0 \Rightarrow f \left(x\right)$ is increasing in $\left(\frac{\pi }{2} , \pi \right) .$