Q. 205.0( 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}


Note: As [.] represents greatest integer function which gives greatest integer less than the number inside [.].


E.g. [1.29] = 1; [-4.65] = -4 ; [9] = 9


[2-h] is just less than 2 say 1.9999 so [1.999] = 1


LHL = (2-0) ×1


LHL = 2 …(2)


Similarly,


RHL =


RHL = {using equation 1}


RHL = (1+0)(2+0) = 2 …(3)


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


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}


Note: As [.] represents greatest integer function which gives greatest integer less than the number inside [.].


E.g. [1.29] = 1 ; [-4.65] = -4 ; [9] = 9


[2-h] is just less than 2 say 1.9999 so [1.999] = 1


LHD =


LHD =


LHD = 1 …(5)


Now,


RHD =


RHD = {using equation 1}


RHD =


RHD =


RHD = 0+3 = 3 …(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


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