Q. 214.5( 2 Votes )

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 = 0


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 = 0 if -



LHL =


LHL = {using equation 1}


As sin (-1/h) is going to be some finite value from -1 to 1 as h0


LHL = 02 × (finite value) = 0


LHL = 0 …(2)


Similarly,


RHL =


RHL = {using equation 1}


As sin (1/h) is going to be some finite value from -1 to 1 as h0


RHL = (0)2(finite value) = 0 …(3)


And, f(0) = 0 {using equation 1} …(4)


From equation 2,3 and 4 we observe that:



f(x) is continuous at x = 0. So we will proceed now to check the differentiability.


Checking for the differentiability:


Now according to above theory-


f(x) is differentiable at x = 0 if -



LHD =


LHD = {using equation 1}


LHD =


As sin (1/h) is going to be some finite value from -1 to 1 as h0


LHD = 0×(some finite value) = 0


LHD = 0 …(5)


Now,


RHD =


RHD = {using equation 1}


RHD =


As sin (1/h) is going to be some finite value from -1 to 1 as h0


RHD = 0×(some finite value) = 0


RHD = 0 …(6)


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


(LHD at x=0) = (RHD at x = 0)


f(x) is differentiable at x = 0


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