(C) Given, $f(x) = \frac{1}{2}[|\sin x| + \sin x]$ for $0 < x \leq 2\pi$. Case $I$: When $0 < x \leq \pi$, $\sin x \geq 0$, so $|\sin x| = \sin x$. Th

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increasing in $\left(0, \frac{\pi}{2}\right)$ and decreasing in $\left(\frac{\pi}{2}, \pi\right)$

(C) Given, $f(x) = \frac{1}{2}[|\sin x| + \sin x]$ for $0 Case $I$: When $0 Thus, $f(x) = \frac{1}{2}[\sin x + \sin x] = \sin x$. The derivative is $f'(x) = \cos x$. For $0 0$, so $f(x)$ is increasing. For $\frac{\pi}{2} Case $II$: When $\pi Thus, $f(x) = \frac{1}{2}[-\sin x + \sin x] = 0$. Therefore, $f(x)$ is increasing in $\left(0, \frac{\pi}{2}\right)$ and decreasing in $\left(\frac{\pi}{2}, \pi\right)$.

Class 12 Mathematics Applications of Derivatives Increasing and Decreasing function

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Define $f(x) = \frac{1}{2}[|\sin x| + \sin x]$, $0 < x \leq 2\pi$. Then, $f$ is

TS EAMCET 2016 All TS EAMCET

A
increasing in $\left(\frac{\pi}{2}, \frac{3\pi}{2}\right)$
B
decreasing in $\left(0, \frac{\pi}{2}\right)$ and increasing in $\left(\frac{\pi}{2}, \pi\right)$
C
increasing in $\left(0, \frac{\pi}{2}\right)$ and decreasing in $\left(\frac{\pi}{2}, \pi\right)$
D
increasing in $\left(0, \frac{\pi}{4}\right)$ and decreasing in $\left(\frac{\pi}{4}, \pi\right)$

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Applications of Derivatives

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Increasing and Decreasing function

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Define $f(x) = \frac{1}{2}[|\sin x| + \sin x]$, $0 < x \leq 2\pi$. Then, $f$ is

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Let $\lambda^{*}$ be the largest value of $\lambda$ for which the function $f_{\lambda}(x) = 4\lambda x^{3} - 36\lambda x^{2} + 36x + 48$ is increasing for all $x \in \mathbb{R}$. Then $f_{\lambda^{*}}(1) + f_{\lambda^{*}}(-1)$ is equal to:

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Let $\lambda^{}$ be the largest value of $\lambda$ for which the function $f_{\lambda}(x) = 4\lambda x^{3} - 36\lambda x^{2} + 36x + 48$ is increasing for all $x \in \mathbb{R}$. Then $f_{\lambda^{}}(1) + f_{\lambda^{*}}(-1)$ is equal to: