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  • Let S be the set of points where the function, <img alt="f(x)=\left | 2-\left | x-3 \right | \right |,x\epsilon R," src="http://entrancecorner.oncodecogs.com/gif.latex?f%28x%29%3D%5Cleft%20%7C%202-%5Cleft%20%7C%20x-3%20%5Cright%20%7C%20%5Cright%20

Let S be the set of points where the function, f(x)=\left | 2-\left | x-3 \right | \right |,x\epsilon R, is not differentiable. Then  \sum_{x\epsilon S}f(f(x))\; is equal to  _______. 
Option: 1 33
Option: 2 10
Option: 3 6
Option: 4 2
 

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Differentiability and Continuity -

 

Consider the function, f(x) = |x|, the modulus function is continuous at x = 0 but it is not differentiable at x= 0. As we see in the graph, at x = 0, it has a sharp edge, hence not differentiable.   

Function fail to Differentiable if

\\\boldsymbol{a.}\text {\;\; both } R f^{\prime}(a) \text { and } L f^{\prime}(a) \text { exist but are not equal, } \\ \mathbf{b.}\text {\;\; either or both } R f^{\prime}(a) \text { and } L f^{\prime}(a) \text { are not finite, and } \\ \mathbf{c.}\text {\;\; either or both } R f^{\prime}(a) \text { and } L f^{\prime}(a) \text { do not exist. }

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\\f(x)=|2-|x-3||\\f(f(x))=|2-||2-|x-3||-3||\\

\\S=\left \{ (1,0),(3,2),(5,0) \right \}\\\sum_{x\in S}f(f(x))=\sum_{x\in S}|2-||2-|x-3||-3||\\=3

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Kuldeep Maurya

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