(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 Application of Derivatives Increasing and Decreasing function

Easy

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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Application of Derivatives

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Increasing and Decreasing function

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TS EAMCET 2016 Paper All TS EAMCET years →

The graph of the function $f(x) = \frac{\cos x}{\cos 2x}$ in the domain $\left(-\frac{\pi}{4}, \frac{\pi}{4}\right)$ is

KVPY 2021

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Let $f(x) = \sin x$ and $g(x) = x$.
Statement-$1$: For $x \in (0, \infty)$,$f(x) \leq g(x)$.
Statement-$2$: For $x \in (0, \infty)$,$f(x) \leq 1$ but as $x \rightarrow \infty$,$g(x) \rightarrow \infty$.

Difficult

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Find the intervals in which the function $f$ given by $f(x)=2 x^{2}-3 x$ is
$(a)$ increasing
$(b)$ decreasing

Medium

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Let $f : R \rightarrow R$ be defined as,
$f(x)=\begin{cases}-55 x, & \text{if } x<-5 \\ 2 x^{3}-3 x^{2}-120 x, & \text{if } -5 \leq x \leq 4 \\ 2 x^{3}-3 x^{2}-36 x-336, & \text{if } x>4 \end{cases}$
Let $A=\{ x \in R : f \text{ is increasing} \}$. Then $A$ is equal to :

DifficultJEE MAIN 2021

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The function $f(x) = ax + b$ is strictly increasing for all real $x$ if

EasyWBJEE 2011

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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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The graph of the function $f(x) = \frac{\cos x}{\cos 2x}$ in the domain $\left(-\frac{\pi}{4}, \frac{\pi}{4}\right)$ is

Let $f(x) = \sin x$ and $g(x) = x$.Statement-$1$: For $x \in (0, \infty)$,$f(x) \leq g(x)$.Statement-$2$: For $x \in (0, \infty)$,$f(x) \leq 1$ but as $x \rightarrow \infty$,$g(x) \rightarrow \infty$.

Find the intervals in which the function $f$ given by $f(x)=2 x^{2}-3 x$ is$(a)$ increasing$(b)$ decreasing

Let $f : R \rightarrow R$ be defined as,$f(x)=\begin{cases}-55 x, & \text{if } x<-5 \\ 2 x^{3}-3 x^{2}-120 x, & \text{if } -5 \leq x \leq 4 \\ 2 x^{3}-3 x^{2}-36 x-336, & \text{if } x>4 \end{cases}$Let $A=\{ x \in R : f \text{ is increasing} \}$. Then $A$ is equal to :

The function $f(x) = ax + b$ is strictly increasing for all real $x$ if

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Let $f(x) = \sin x$ and $g(x) = x$.
Statement-$1$: For $x \in (0, \infty)$,$f(x) \leq g(x)$.
Statement-$2$: For $x \in (0, \infty)$,$f(x) \leq 1$ but as $x \rightarrow \infty$,$g(x) \rightarrow \infty$.

Find the intervals in which the function $f$ given by $f(x)=2 x^{2}-3 x$ is
$(a)$ increasing
$(b)$ decreasing

Let $f : R \rightarrow R$ be defined as,
$f(x)=\begin{cases}-55 x, & \text{if } x<-5 \\ 2 x^{3}-3 x^{2}-120 x, & \text{if } -5 \leq x \leq 4 \\ 2 x^{3}-3 x^{2}-36 x-336, & \text{if } x>4 \end{cases}$
Let $A=\{ x \in R : f \text{ is increasing} \}$. Then $A$ is equal to :