Let f(x) = (x + |x|) |x|. Then, for all x: (a) f is continuous. (… - Vidyadip

MCQ

Let $f(x) = (x + |x|) |x|$. Then, for all $x:$
$(a) \ f$ is continuous.
$(b)\ f$ is differentiable for some $x$
$(c)\ f\ '$ is continuous.
$(d)\ f\ ''$ is continuous.

✓
$a$ and $c$
B
$b $ and $c$
C
$a$ and $d$
D
$b$ and $d$

✓

Answer

Correct option: A.

$a$ and $c$

$\text{f(x)}=(\text{x}+|\text{x}|)|\text{x}|$
$\Rightarrow\text{f(x)}=2\text{x}^2,\text{ x} > 0$
$=0,\text{ x} < 0$
$\lim\limits_{\text{x}\rightarrow0}2\text{x}^2=0$
Function is continuous at $x = 0.$
Also, differentiable at $x = 0$ as it is polynomial function.

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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $f(x) = \frac{x}{{x - 1}} = \frac{1}{y}$, then $f(y) = $

Given $A=\left[\begin{array}{ccc}\sqrt{3} & 1 & -1 \\ 2 & 3 & 0\end{array}\right]$ and $B=\left[\begin{array}{ccc}2 & \sqrt{5} & 1 \\ -2 & 3 & \frac{1}{2}\end{array}\right],$ find $A+B$ $=$ ........

If a, b, c are distinct, then the value of x satisfying $\begin{vmatrix}0&\text{x}^2-\text{a}&\text{x}^3-\text{b}\\\text{x}^2+\text{a}&0&\text{x}^2+\text{c}\\\text{x}^4+\text{b}&\text{x}-\text{c}&0\end{vmatrix}=0$ is:

If $\tan ^{-1} \frac{x-1}{x-2}+\tan ^{-1} \frac{x+1}{x+2}=\frac{\pi}{4},$ then find the value of $\mathrm{x}$

The value of $\int\limits^\pi_{-\pi}\sin^3\text{x}\cos^2\text{x}\text{ dx}$ is :

The signum function, $f: R \rightarrow R$ is given by $f(x)=\left\{\begin{array}{ll}1, & x>0 \\ 0, & x=0 \\ -1, & x<0\end{array}\right.$ is

If $f(x) = \left\{ {\begin{array}{*{20}{c}}
{x\left[ x \right],\,\,\,\,\,\,\,\,\,\,\,\,\,}&{0 \le x < 2}\\
{\left( {x - 1} \right)\left[ x \right]\,,\,\,\,}&{2 \le x \le 4}
\end{array}} \right.,$ where $[.]$ denotes greatest integer function, then

The objective function $Z = 4x + 3y$ can be maximised subjected to the constraints $3x + 4y \leq 24, 8x + 6y \leq 48, x \leq 5, y \leq 6, x, y \geq 0$

Let $f_n=\int \limits_0^{\frac{\pi}{2}}\left(\sum \limits_{k=1}^n \sin ^{k-1} x\right)\left(\sum \limits_{k=1}^n(2 k-1) \sin ^{k-1} x\right) \cos x$ $d x, n \in N$. Then $f_{21}-f_{20}$ is equal to $...........$.

A coin is tossed $n$ times. The probability of geting at least once is greater than $0.8$. Then, the least value of $n,$ is: