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- (i) If $f'$ is positive on an open interval $I$. Then $f$ is strictly increasing on $I$
- (ii) If $f'$ is negative on an open interval $I$, then $f$ is strictly decreasing on $I$

$f(x)= x \sin x $ on $\bigg[0, \large\frac{\pi}{4}\bigg]$

Step 1:

$f'(x)=\sin x +x \cos x \qquad f'(0)=0$ and

Step 2:

$f'(x) >0$ for $ x \in \bigg[0, \large\frac{\pi}{4}\bigg]$

$\therefore \;f'(x) \leq 0\;on \;\bigg[0, \large\frac{\pi}{4}\bigg]$

It is increasing on $\bigg[0,\large\frac{\pi}{4}\bigg]$

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