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Which of the following function are increasing or decreasing on the interval given? \[\] $x\sin x$ on$ [0 , \large\frac{\pi}{4}]$

Note: This is part 5th of a 5 part question, split as 5 separate questions here.

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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 $\bigg[0,\large\frac{\pi}{4}\bigg]$
answered Jul 30, 2013 by meena.p

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