Show that the function f(x)=x^ 100 + x-1 is increasing on the int… - Vidyadip

Question

Show that the function $f(x)=x^{100}+\sin x-1$ is increasing on the interval $\left(\frac{\pi}{2}, \pi\right)$

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Answer

Given interval : $x \in(\pi / 2, \pi)$
$\Rightarrow \pi / 2 < x<\pi$
$x^{99}>1$
$100 x^{99}>100$
Again, $x \in(\pi / 2, \pi) \Rightarrow-1<\cos x<0 \Rightarrow 0>\cos x>-1$
$100 x^{99}>100$ and $\cos x>-1$
$100 x^{99}+\cos x>100-1=99$
$100 x^{99}+\cos x>0$
$f^{\prime}(x)>0$
Thus $f(x)$ is increasing on $(\pi / 2, \pi)$

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