Q. 225.0( 1 Vote )

Examine the differentiability of f, where f is defined by


Answer :

Given,


…(1)


We need to check whether f(x) is continuous and differentiable at x = 2


A function f(x) is said to be continuous at x = c if,


Left hand limit(LHL at x = c) = Right hand limit(RHL at x = c) = f(c).


Mathematically we can represent it as-



Where h is a very small number very close to 0 (h0)


And a function is said to be differentiable at x = c if it is continuous there and


Left hand derivative(LHD at x = c) = Right hand derivative(RHD at x = c) = f(c).


Mathematically we can represent it as-




Finally we can state that for a function to be differentiable at x = c



Checking for the continuity:


Now according to above theory-


f(x) is continuous at x = 2 if -



LHL =


LHL = {using equation 1}


LHL =


LHL = (3-h) = 3


LHL = 3 …(2)


Similarly,


RHL =


RHL = {using equation 1}


RHL =


RHL = 3+0 = 3 …(3)


And, f(2) = 1 + 2 = 3 {using equation 1} …(4)


From equation 2,3 and 4 we observe that:



f(x) is continuous at x = 2. So we will proceed now to check the differentiability.


Checking for the differentiability:


Now according to above theory-


f(x) is differentiable at x = 2 if -



LHD =


LHD = {using equation 1}


LHD =


LHD = 1 …(5)


Now,


RHD =


RHD = {using equation 1}


RHD =


RHD = -1 …(6)


Clearly from equation 5 and 6,we can conclude that-


(LHD at x=2) ≠ (RHD at x = 2)


f(x) is not differentiable at x = 2


Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Super 10 Question: Check Your Knowledge of Maxima & Minima (Quiz)Super 10 Question: Check Your Knowledge of Maxima & Minima (Quiz)Super 10 Question: Check Your Knowledge of Maxima & Minima (Quiz)45 mins
Maxima & Minima in an intervalMaxima & Minima in an intervalMaxima & Minima in an interval60 mins
Connection B/w Continuity & DifferentiabilityConnection B/w Continuity & DifferentiabilityConnection B/w Continuity & Differentiability59 mins
Questions Based on Maxima & MinimaQuestions Based on Maxima & MinimaQuestions Based on Maxima & Minima57 mins
Questions based on Maxima & Minima in an intervalQuestions based on Maxima & Minima in an intervalQuestions based on Maxima & Minima in an interval59 mins
Check your Knowlege of Maxima & Minima ( Challenging Quiz)Check your Knowlege of Maxima & Minima ( Challenging Quiz)Check your Knowlege of Maxima & Minima ( Challenging Quiz)60 mins
Problems Based on L-Hospital Rule (Quiz)Problems Based on L-Hospital Rule (Quiz)Problems Based on L-Hospital Rule (Quiz)0 mins
When does a Maxima or Minima occur?When does a Maxima or Minima occur?When does a Maxima or Minima occur?48 mins
Interactive Quiz | Differentiability by using first principleInteractive Quiz | Differentiability by using first principleInteractive Quiz | Differentiability by using first principle59 mins
Interactive Quiz on LimitsInteractive Quiz on LimitsInteractive Quiz on Limits67 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses