Download App

CONTINUITY AND DIFFERENTIABILITY — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSCONTINUITY AND DIFFERENTIABILITY3 Marks
Question
Prove that the greatest integer function defined by
$\text{f(x)} = [\text{x} ], 0 < \text{x} < 3$
is not differentiable at x = 1 and x = 2.
✓

Answer

Given: $\text{f(x)} = [\text{x} ], 0 < \text{x} < 3$
$\text{R}\text{f}{'}(1) = ^{\ \ \text{lim}}_{\text{h}\rightarrow\text{0}}\frac{\text{f(1 + h)}-\text{f}(1)}{\text{h}} = ^{\ \ \text{lim}}_{\text{h}\rightarrow\text{0}}\frac{|1 + \text{h} |-1}{\text{h}}$$ = ^{\ \ \text{lim}}_{\text{h}\rightarrow\text{0}}\frac{1-1}{\text{h}}= ^{\ \ \text{lim}}_{\text{h}\rightarrow\text{0}}\frac{0}{\text{h}}=0$
$\text{And}\ \text{L}\text{f}{'}(1) = ^{\ \ \text{lim}}_{\text{h}\rightarrow\text{0}}\frac{\text{f(1 - h)}-\text{f}(1)}{-\text{h}} = ^{\ \ \text{lim}}_{\text{h}\rightarrow\text{0}}\frac{|1 - \text{h}| - 1}{-\text{h}}$$ = ^{\ \ \text{lim}}_{\text{h}\rightarrow\text{0}}\frac{|1-\text{h}|-1}{\text{h}}= ^{\ \ \text{lim}}_{\text{h}\rightarrow\text{0}}\frac{0-1}{-\text{h}}=\infty$
Since $\text{R }\text{f}{'}(1)\neq \text{L}{\text{f}}{'}(1)$
Therefore, f(x) = [x] is not differentiable at x =1.
Similarly f(x) = [x] is not differentiable at x = 2.

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 "3 Marks Question" 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 side of a square sheet is increasing at the rate of 4cm per minute. At what rate is the area increasing when the side is 8cm long?
→
Sum the following siries:
$\tan^{-1}\frac{1}{3}+\tan^{-1}\frac{2}{9}+\tan^{-1}\frac{4}{33}+...+\tan^{-1}\frac{2^{\text{n}-1}}{1+2^{2\text{n}-1}}$
→
$\text{If}\ \vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\ \vec{\text{b}}=2\hat{\text{i}}-\hat{\text{j}}+3\hat{\text{k}}$ and $\vec{\text{c}}=\hat{\text{i}}-2\hat{\text{j}}+\hat{\text{k}},$ find a unit vector parallel to the vector $2\vec{\text{a}}-\vec{\text{b}}+3\vec{\text{c}}.$
→
Solve the Linear Programming Problem graphically : Maximize $Z = 3x + 4y$ Subject to $2 x+2 y \leq 80\  2 x+4 y \leq 120$
→
Solve the following differential equations:$2\text{x}\frac{\text{dy}}{\text{dx}}=5\text{y},\text{y}(1)=1$
→
Classify the following functions as injection, surjection or bijection: $f : Z \rightarrow Z,$ defined by $f(x) = x^2 + x$
→
Tickets are numbered from 1 to 10. Two tickets are drawn one after the other at random. Find the probability that the number on one of the tickets is a multiple of 5 and on the other a multiple of 4.
→
Evaluate the following integrals:
$\int3\sqrt{\cos^2\text{x}}\sin\text{x }\text{dx}$
→
Show that the function $\text{f}(\text{x}) = |\text{x} - 3 |,\text{x}\in|\text{R},$is continuous but not differentiable at x = 3.
→
Using determinants, find the area of the triangle whose vertices are $(1, 4), (2, 3)$ and $(-5, -3).$ Are the given points collinear$?$
→