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Find the intervals on which $f$ is increasing or decreasing. $f(x)=\sin^{4}x+\cos^{4}x $ in $ [0 , \large\frac{\pi}{2}]$

Note: This is part 6 of a 6 part question, split as 6 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 $I$
$f(x) =\sin ^4 x +\cos ^4 x \qquad x \in [0, \pi]$
Step 1:
$f'(x)=4 \sin ^3 x \cos x -4 \cos ^3 x \sin x$
$\qquad=4 \sin x \cos x (\sin ^2 x -\cos ^2 x)$
Step 2:
$f'(x)=0$ for $x \in (0, \large\frac{\pi}{2})$
When $\sin ^2 x -\cos ^2 x =0\qquad (\sin x, \cos x \neq 0)$
$\therefore x=\large \frac{\pi}{4}$
In the interval $(0,\large\frac{\pi}{4}) $$\cos x > \sin x$ $ \therefore f'(x) <0$
In the interval $(\large\frac{\pi}{4},\frac{\pi}{2}) $$\sin x > \cos x$ $ \therefore f'(x) >0$
$f(x)$ is decreasing in $[0,\large\frac{\pi}{4}]$
$f(x)$ is increasing in $[\large\frac{\pi}{4},\frac{\pi}{2}]$
answered Jul 30, 2013 by meena.p
 

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