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Show that the Function F(X) = Sin (2x + π/4) is Decreasing on (3π/8, 5π/8) ? - CBSE (Arts) Class 12 - Mathematics

ConceptIncreasing and Decreasing Functions

Question

Show that the function f(x) = sin (2x + π/4) is decreasing on (3π/8, 5π/8) ?

Solution

$f\left( x \right) = \sin \left( 2x + \frac{\pi}{4} \right)$

$f'\left( x \right) = 2 \cos \left( 2x + \frac{\pi}{4} \right)$

$\text { Here, }$

$\frac{3\pi}{8} < x < \frac{5\pi}{8}$

$\Rightarrow \frac{3\pi}{4} < 2x < \frac{5\pi}{4}$

$\Rightarrow \pi < 2x + \frac{\pi}{4} < \frac{3\pi}{2}$

$\Rightarrow \ cos \left( 2x + \frac{\pi}{4} \right) < 0 \left[ \because \text { Cos function is negative inthird quadrant } \right]$

$\Rightarrow 2 \cos \left( 2x + \frac{\pi}{4} \right) < 0$

$\Rightarrow f'\left( x \right) < 0, \forall x \in \left( \frac{3\pi}{8}, \frac{5\pi}{8} \right)$

$\text { So },f\left( x \right) \text { is decreasing on }\left( \frac{3\pi}{8}, \frac{5\pi}{8} \right).$

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Solution Show that the Function F(X) = Sin (2x + π/4) is Decreasing on (3π/8, 5π/8) ? Concept: Increasing and Decreasing Functions.
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