Which of the following functions is differentiable at x = 0 - Vidyadip

MCQ

Which of the following functions is differentiable at $x = 0$

A
$\cos (|x|)\, + |x|$
B
$\cos (|x|)\, - |x|$
C
$\sin (|x|)\, + |x|$
✓
$\sin (|x|)\, - |x|$

✓

Answer

Correct option: D.

$\sin (|x|)\, - |x|$

d
(d) $\cos |x|\, = \cos x$ which is differentiable everywhere and $|x|$ is not differentiable at $x = 0$. So, both $\cos (|x|) \pm |x|$ are not differentiable at $x = 0$.
For $x < 0,\,\sin (|x|)\, + |x|\, = \sin ( - x) - x = - \sin x - x$
For $x > 0,\,\sin (|x|)\, + |x|\, = \sin x + x$.
$\therefore$ Its $LHD$ $x = 0$ is ${[ - \cos x - 1]_{x = 0}} = - 2$
and its $RHD$ at $x = 0$ is ${[\cos x + 1]_{x = 0}} = 2$.
$\therefore$ $\sin (|x|)\, + |x|$ is not differentiable at $x = 0$.
For $x < 0,\,\sin (|x|)\,\, - |x|\, = \sin ( - x) - ( - x) = - \sin x + x$
For $x > 0,\,\sin (|x|)\, - |x|\, = \sin x - x$
$\therefore$ its $LHD$ at $x = 0$ is ${[ - \cos x + 1]_{x = 0}} = 0$ and its $RHD$ at $x = 0$ is ${[\cos x - 1]_{x = 0}} = 0$
$\therefore$ $\left( {\sin (|x|)\,\, - |x|} \right)$ is differentiable at $x = 0$.

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 5. continuity and differentiation→5. continuity and differentiation — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\vec{a} \times \vec{b}=\hat{i}-\hat{j}$ then the value of $\vec{b} \times \vec{a}$

The function f(x) = x²e^-x is monotonic increasing when:

$\text{x}\in\text{R}-[0,2]$
$0<\text{x}<2$
$2<\text{x}<\infty$
$\text{x}<0$

Let the sets $A$ and $B$ denote the domain and range respectively of the function $f(x)=\frac{1}{\sqrt{\lceil x\rceil-x}}$ where $\lceil x \rceil$ denotes the smallest integer greater than or equal to $x$. Then among the statements

$( S 1): A \cap B =(1, \infty)-N$ and

$( S 2): A \cup B=(1, \infty)$

The solution of the given differential equation $\frac{{dy}}{{dx}} + 2xy = y$ is

For $a>0,$ let the curves $C_{1}: y^{2}=a x$ and $\mathrm{C}_{2}: \mathrm{x}^{2}=$ ay intersect at origin $\mathrm{O}$ and a point $\mathrm{P}$ Let the line $\mathrm{x}=\mathrm{b}(0<\mathrm{b}<\mathrm{a})$ intersect the chord $OP$ and the $\mathrm{x}$ -axis at points $\mathrm{Q}$ and $\mathrm{R}$, respectively. If the line $x=b$ bisects the area bounded by the curves, $\mathrm{C}_{1}$ and $\mathrm{C}_{2},$ and the area of $\Delta \mathrm{OQR}=\frac{1}{2},$ then '$a$' satisfies the equation

Let $I_{n}=\int_{1}^{e} x^{19}(\log |x|)^{n} d x,$ where $n \in N$. If $(20) I _{10}=\alpha I _{9}+\beta I _{8},$ for natural numbers $\alpha$ and $\beta$, then $\alpha-\beta$ equal to ..... .

The value of $\int\frac{\sin\text{x}+\cos\text{x}}{\sqrt{1-\sin2\text{x}}}\text{ dx}$ is equal to:

$\sqrt{\sin2\text{x}}+\text{C}$
$\sqrt{\cos2\text{x}}+\text{C}$
$\pm(\sin\text{x}-\cos\text{x})+\text{C}$
$\pm\log(\sin\text{x}-\cos\text{x})+\text{C}$

Let for any three distinct consecutive terms $a, b, c$ of an $A.P,$ the lines $a x+b y+c=0$ be concurrent at the point $\mathrm{P}$ and $\mathrm{Q}(\alpha, \beta)$ be a point such that the system of equations $ x+y+z=6, $ $ 2 x+5 y+\alpha z=\beta$ and $x+2 y+3 z=4$, has infinitely many solutions. Then $(P Q)^2$ is equal to________.

The solution of $y' = 1 + x + {y^2} + x{y^2}$, $y(0) = 0$ is

If $A B C D$ is a parallelogram and $A C$ and $B D$ are its diagonals, then $\overrightarrow{A C}+\overrightarrow{B D}$ is: