If the function f(x)= -x 2 + defined on [- 3 , 3 ] is : - Vidyadip

MCQ

If the function $\text{f}(\text{x})=\frac{-\text{x}}{2}+\sin\text{x}$ defined on $\Big[\frac{-\pi}{3},\frac{\pi}{3}\Big]$ is :

✓
Increasing.
B
Decreasing.
C
Constant.
D
None of these.

✓

Answer

Correct option: A.

Increasing.

$\text{f}(\text{x})=\frac{-\text{x}}{2}+\sin\text{x}$ defined on $\Big[\frac{-\pi}{3},\frac{\pi}{3}\Big]$
$\therefore\ \text{f}\ '(\text{x})=\frac{-1}{2}+\cos\text{x}$
$\Rightarrow\text{f}\ '(\text{x})\geq0,\forall\ \text{x}\in\Big[\frac{-\pi}{3},\frac{\pi}{3}\Big]$
$\Big[\because\ \text{for }\text{x}\in\Big[\frac{-\pi}{3},\frac{-\pi}{3}\Big],\cos\geq\frac{1}{2}\Big]$
Hence, the given function is increasing.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from Application of Derivatives→Application of Derivatives — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $f\left( x \right) + 2f\left( {\frac{1}{x}} \right) = 3x,x \ne 0$ and $S = \left\{ {x \in R:f\left( x \right) = f\left( { - x} \right)} \right\}$;then $S :$

Which of the given qualities is a vector:

The value of $\int_0^{{{\sin }^2}x} {{{\sin }^{ - 1}}\sqrt t \,dt + \int_0^{{{\cos }^2}x} {{{\cos }^{ - 1}}\sqrt t \,dt} } $ is

$\int_{}^{} {x\sin x{{\sec }^3}x\,dx = } $

$f(x)=x^x$ has a stationary point at

For any positive integer $n$, define $f_n:(0, \infty) \rightarrow R$ as

$f_n(x)=\sum_{j=1}^n \tan ^{-1}\left(\frac{1}{1+(x+j)(x+j-1)}\right) \text { for all } x \in(0, \infty)$

(Here, the inverse trigonometric function $\tan ^{-1} x$ assumes values in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ )

Then, which of the following statement(s) is (are) TRUE?

$(A)$ $\sum_{ j =1}^5 \tan ^2\left( f _{ j }(0)\right)=55$

$(B)$ $\sum_{ j =1}^{10}\left(1+ f _{ j }^{\prime}(0)\right) \sec ^2\left( f _{ j }(0)\right)=10$

$(C)$ For any fixed positive integer $n$, $\lim _{x \rightarrow \infty} \tan \left(f_n(x)\right)=\frac{1}{n}$

$(D)$ For any fixed positive integer $n, \lim _{x \rightarrow \infty} \sec ^2\left(f_n(x)\right)=1$

Choose the correct answer from the given four options : Let the $f : R \rightarrow R$ be defined by $\text{f(x)}=2\text{x}+\cos\text{x}$ then $f :$

If $(a, b)$ be the orthocentre of the triangle whose vertices are $(1,2),(2,3)$ and $(3,1)$, and $I_1=\int_{\mathrm{a}}^{\mathrm{b}} \mathrm{x} \sin \left(4 \mathrm{x}-\mathrm{x}^2\right) \mathrm{dx}, \mathrm{I}_2=\int_{\mathrm{a}}^{\mathrm{b}} \sin \left(4 \mathrm{x}-\mathrm{x}^2\right) \mathrm{dx}$ , then $36 \frac{\mathrm{I}_1}{\mathrm{I}_2}$ is equal to :

If a line makes angle $\alpha,\beta$ and $\gamma$ with the axes respectively, then $\cos2\alpha+\cos2\beta+\cos2\gamma=$

$\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)+\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)+\tan ^{-1}\left(\tan \frac{3 \pi}{4}\right) \quad$ is equal to