On the interval left( 0, frac{pi}{2} right),t | vedclass

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(A) Let $f(x) = \log(\sin x)$.To determine if the function is increasing or decreasing,we find its derivative $f'(x)$.$f'(x) = \frac{d}{dx}(\log(\sin

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Increasing

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(A) Let $f(x) = \log(\sin x)$.To determine if the function is increasing or decreasing,we find its derivative $f'(x)$.$f'(x) = \frac{d}{dx}(\log(\sin x)) = \frac{1}{\sin x} \cdot \cos x = \cot x$.In the interval $\left( 0, \frac{\pi}{2} \right)$,the value of $\cot x$ is always positive (since both $\sin x$ and $\cos x$ are positive in the first quadrant).Since $f'(x) > 0$ for all $x \in \left( 0, \frac{\pi}{2} \right)$,the function $f(x) = \log(\sin x)$ is strictly increasing on this interval.Therefore,the correct option is $A$.

On the interval $\left( 0, \frac{\pi}{2} \right)$,the function $\log(\sin x)$ is

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