The function f(x)= (x|x- |) 4+|x|^2 , where[.] denotes the greate… - Vidyadip

Question

The function $\text{f(x)}=\frac{\sin(\text{x}|\text{x}-\pi|)}{4+|\text{x}|^2},$ where[.] denotes the greatest integer function, is:

Continuous as well as differentiable for all $\text{x}\in\text{R}$
Continuous for all x but differentiable at some x
Differentiable for all x but not continuous at some x
None of these.

✓

Answer

Continuous as well as differentiable for all $\text{x}\in\text{R}$

Solution:
Here,
$\text{f(x)}=\frac{\sin(\text{x}|\text{x}-\pi|)}{4+|\text{x}|^2}$
Since, we know that $\pi(\text{x}-\pi)=\text{n}\pi$ and $\sin\text{n}\pi=0.$
$\because4+\text{x}[\text{x}]^2\neq0$
$\therefore\text{f(x)}=0$ for all x
Thus, f(x) is a constant function and it is continuous and differentible everywhere.

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 CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

The equation $2\cos^{-1}\text{x}+\sin^{-1}\text{x}=\frac{11\pi}{6}$ has:

No solution.
Only one solution.
Two solutions.
Three solutions.

The Cartesian equation of the line passing through the point $(1,-3,2)$ and parallel to the line $\vec{r}=(2+\lambda) \hat{i}+\lambda \hat{j}+(2 \lambda-1) \hat{k}$ is

Choose the correct answer from given four options in each of the Exercise:
The maximum value of $\Delta=\begin{vmatrix}1 & 1&1 \\1 &1+\sin\theta&1\\1+\cos\theta &1&1\end{vmatrix}$ is ($\theta$ is real number):

$\frac{1}{2}$
$\frac{\sqrt{3}}{2}$
$\sqrt{2}$
$\frac{2\sqrt{3}}{4}$

Let X denote the number of times heads occur in n tosses of a fair coin. If P(X = 4), P(X = 5) and P(X = 6) are in AP, the value of n is:

7, 14
10, 14
12, 7
14, 12

Choose the correct answers from the given four options:
The function $\text{f(x)}=\cot\text{x}$ is discontinuous on the set

$\big\{\text{x}=\text{n}\pi:\text{n}\in\text{Z}\big\}$
$\big\{\text{x}=2\text{n}\pi:\text{n}\in\text{Z}\big\}$
$\Big\{\text{x}=(2\text{n}+1)\frac{\pi}{2};\text{n}\in\text{Z}\Big\}$
$\Big\{\text{x}=\frac{\text{n}\pi}{2};\text{n}\in\text{Z}\Big\}$

$\int\left(\sin ^{-1} x+\cos ^{-1} x\right) d x$ is equal to :

It is given that $X\left[\begin{array}{cc}3 & 2 \\ 1 & -1\end{array}\right]=\left[\begin{array}{ll}4 & 1 \\ 2 & 3\end{array}\right]$. Then matrix $X$ is :

If $\sin^{-1}(\text{x}^2-7\text{x}+12)=\text{n}\pi,\forall\text{ n }\epsilon\text{ I},$ then x =

-2
4
-3
5

If $\displaystyle \text{a}_{\text{ij}}=0\left (\text{i}\neq \text{j} \right )$ and $\displaystyle \text{a}_{\text{ij}}=1\left (\text{i}= \text{j} \right )$ then the matrix $\text{A}=\displaystyle \left [\text{a}_{\text{ij}} \right ]_{\text{n}\times\text{n}}$ is a _____ matrix:

Null
Identity
Scalar
Triangular

In which of the function is one-one-