Share

Books Shortlist
Your shortlist is empty

Solution for Find the Interval in Which F(X) is Increasing Or Decreasing F(X) = Sinx(1 + Cosx), 0 < X < π 2 ? - CBSE (Science) Class 12 - Mathematics

ConceptIncreasing and Decreasing Functions

Question

Find the interval in which f(x) is increasing or decreasing f(x) = sinx(1 + cosx), 0 < x < $\frac{\pi}{2}$ ?

Solution

$f\left( x \right) = \sin x\left( 1 + \cos x \right), 0 < x < \frac{\pi}{2}$

$\Rightarrow f\left( x \right) = \sin x + \sin x\cos x$

$\Rightarrow f'\left( x \right) = \cos x + \sin x\left( - \sin x \right) + \cos x\cos x$

$\Rightarrow f'\left( x \right) = \cos x - \sin^2 x + \cos^2 x$

$\Rightarrow f'\left( x \right) = \cos x + \cos^2 x - 1 + \cos^2 x$

$\Rightarrow f'\left( x \right) = 2 \cos^2 x + \cos x - 1$

$\Rightarrow f'\left( x \right) = 2 \cos^2 x + 2\cos x - \cos x - 1$

$\Rightarrow f'\left( x \right) = 2\cos x\left( \cos x + 1 \right) - 1\left( \cos x + 1 \right)$

$\Rightarrow f'\left( x \right) = \left( 2\cos x - 1 \right)\left( \cos x + 1 \right)$

$\text { For } f\left( x \right) \text { to be increasing, we must have }$

$f'\left( x \right) > 0$

$\Rightarrow \left( 2\cos x - 1 \right)\left( \cos x + 1 \right) > 0$

$\text { This is only possible when}$

$\left( 2\cos x - 1 \right) < 0 \text { and } \left( \cos x + 1 \right) > 0$

$\Rightarrow \left( 2\cos x - 1 \right) < 0 \text { and } \left( \cos x + 1 \right) > 0$

$\Rightarrow \cos x < \frac{1}{2} \text { and} \cos x > - 1$

$\Rightarrow x \in \left( \frac{\pi}{3}, \frac{\pi}{2} \right)\text { and } x \in \left( 0, \frac{\pi}{2} \right)$

$\text { So,} x \in \left( \frac{\pi}{3}, \frac{\pi}{2}\right)$

$\therefore f\left( x \right) \text { is decreasing on } \left( \frac{\pi}{3}, \frac{\pi}{2} \right) .$

Is there an error in this question or solution?

Video TutorialsVIEW ALL [3]

Solution Find the Interval in Which F(X) is Increasing Or Decreasing F(X) = Sinx(1 + Cosx), 0 < X < π 2 ? Concept: Increasing and Decreasing Functions.
S