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The function f(x) = 1/(1+|x|) is____

(A) Differntiable and Continuous
(B) Not Differntiable and Not Continuous
(C) Continuous but not Differntiable

I know this is continuous everywhere because |X| will always be a positive value, how to check if it's differntiable or not?
asked in Calculus by Veteran (14.6k points)   | 306 views
continuous but not differentiable?

@vijay find left and right derivative...one will be -1 and other 1 means nt differentiable!!

@Sudsho
yes correct RHD=-1 and LHD=1 hence not differntiable

1 Answer

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Let us define the function as we have |x| here ..

So

f(x)    =    1 / (1 + x)   , if x >= 0

         =    1 / (1 - x)   , if x <= 0

For continuity , we check at x = 0 only , so 

LHL = RHL  = f(0)  =  1

f' (x)   =   - 1 / (1 + x)2  , if x >= 0

         =   -1 / (1 - x)2  * (-1)  ,  

         =    1 / (1 - x)2     , if x <= 0

So Left derivative  at x = 0  :  1

 Right derivative  at x = 0 :  -1

So as Left derivative !=  Right derivative here

Hence  f(x) is not differentiable at x  =  0

Hence the given function is continuous but not differentiable due to x = 0..

answered by Veteran (66.5k points)  
Check the function definition.
This is a 12th class question.. I can bet my solution to be correct @ManojK..

No problem whatsoever..

actually he meant ur conditions are a bit wrong...x>=0 x<=0...on x=0 fn is 1....thats it nthing else;

Here we can write like this also..No problem @sudsho..


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