If f(x)=|x-a| (x), where (x) is continuous function, then: - Vidyadip

MCQ

If $\text{f(x)}=|\text{x}-\text{a}|\ \phi\ (\text{x}),$ where $\phi(\text{x})$ is continuous function, then:

A
$\text{f}'(\text{a}^+)=\phi(\text{a})$
✓
$\text{f}'(\text{a}^-)=-\phi(\text{a})$
C
$\text{f}'(\text{a}^+)=\text{f}'(\text{a}^-)$
D
None of these

✓

Answer

Correct option: B.

$\text{f}'(\text{a}^-)=-\phi(\text{a})$

Given that $\text{f(x)}=|\text{x}-\text{a}|\ \phi\ (\text{x}),$ where $\phi(\text{x})$ continuous function.

$|\text{x}-\text{a}|\Rightarrow\text{x}-\text{a}$ if $\text{x}-\text{a}>0$

$|\text{x}-\text{a}|\Rightarrow-(\text{x}-\text{a})$ if $\text{x}-\text{a}<0$

By definition of continuity,

$\text{f}'(\text{a})= \lim\limits_{\text{h}\rightarrow0}\frac{\text{f}(\text{a}+\text{h})-\text{f(a)}}{\text{h}}$

Hence, $\text{f}(\text{a}^+)=\phi(\text{x})$ and $\text{f}'(\text{a}^-)=-\phi(\text{x})$

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→Continuity — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $y=\log \left(\frac{1-x^2}{1+x^2}\right)$, then $\frac{d y}{d x}$ is equal to

An experiment succeeds twice as often as it fails. The probability of at least $5$ successes in the six trials of this experiment is

The range of the function,

$\mathrm{f}(\mathrm{x})=\log _{\sqrt{5}}(3+\cos \left(\frac{3 \pi}{4}+\mathrm{x}\right)+\cos \left(\frac{\pi}{4}+\mathrm{x}\right)+\cos \left(\frac{\pi}{4}-\mathrm{x}\right)$

$-\cos \left(\frac{3 \pi}{4}-\mathrm{x}\right))$ is :

For the area bounded by the curve $y = ax,$ the line $x = 2$ and $x -$ axis to be $2 \ sq.$ units, the value of a must be equal to:

Choose the correct answer in Exercise$:\ $The value of the integral $\int^{1}_{\frac{1}{3}}\frac{(\text{x}-\text{x}^{3})^{\frac{1}{3}}}{\text{x}^{4}}\text{dx}\ \text{is}$

If ${a^2}{x^4} + {b^2}{y^4} = {c^6},$ then maximum value of $xy $ is

Let a line passing through the point $(-1,2,3)$ intersect the lines $L_1: \frac{x-1}{3}=\frac{y-2}{2}=\frac{z+1}{-2}$ at $\mathrm{M}(\alpha, \beta, \gamma)$ and $\mathrm{L}_2: \frac{\mathrm{x}+2}{-3}=\frac{\mathrm{y}-2}{-2}=\frac{\mathrm{z}-1}{4}$ at $\mathrm{N}(\mathrm{a}, \mathrm{b}$, c). Then the value of $\frac{(\alpha+\beta+\gamma)^2}{(a+b+c)^2}$ equals

Which of the following is an essential condition in a situation for linear programming to be useful?

If $\text{P(A)}=\frac{7}{13},\text{P(B)}=\frac{9}{13}$ and $\text{P}(\text{A}\cap\text{B})=\frac{4}{13}.$ then, $\text{P}(\overline{\text{A}}|\text{B})=$

The number of real values of $\lambda $ for which the system of linear equations $2x + 4y - \lambda z = 0$ ;$4x + \lambda y + 2z = 0$ ; $\lambda x + 2y+ 2z = 0$ has infinitely many solutions, is