Show that the function given by f(x)= x is neither increasing nor… - Vidyadip

Question

Show that the function given by $f(x)=\sin x$ is neither increasing nor decreasing in $(0, \pi)$

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Answer

The function is $f(x)=\sin x$
Then, $f^{\prime}(x)=\cos x$
Since for each $x \in\left(0, \frac{\pi}{2}\right), \cos x >0$, we have $f^{\prime}(x)>0$
Therefore, $f$ is strictly increasing in $\left(0, \frac{\pi}{2}\right)$.
Now, The function is $f ( x )=\sin x$
Then, $f^{\prime}(x)=\cos x$
Since, for each $x \in\left(\frac{\pi}{2}, \pi\right), \cos x <0$, we have $f^{\prime}(x)<0$
Therefore, $f$ is strictly decreasing in $\left(\frac{\pi}{2}, \pi\right) \ldots . .(2)$
From (1) and (2),
It is clear that f is neither increasing nor decreasing in $(0, \pi)$.

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