Determine whether f(x)=- 2 + is a increasing or decreasing on (- … - Vidyadip

Question

Determine whether $\text{f}(\text{x})=\frac{-\pi}{2}+\sin\text{x}$ is a increasing or decreasing on $\Big(\frac{-\pi}{3},\frac{\pi}{3}\Big).$

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Answer

$\text{f}(\text{x})=\frac{-\pi}{2}+\sin\text{x}$
$\text{f}'(\text{x})=\frac{-1}{2}+\cos\text{x}$
Here,
$\frac{-\pi}{3}<\text{x}<\frac{\pi}{3}$
$\Rightarrow\cos\text{x}>\frac{1}{2}$
$\Rightarrow\frac{-1}{2}+\cos\text{x}>0$
$\Rightarrow\text{f}'(\text{x})>0,\forall\ \text{x}\in\Big(\frac{-\pi}{3},\frac{\pi}{3}\Big)$
So, f(x) is increasing on $\Big(\frac{-\pi}{3},\frac{\pi}{3}\Big).$

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