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The function y=|2-3x| is not differential at x=2/3, please prove it.
asked in Calculus by Veteran (14.4k points)   | 99 views
thanks pavan!
habib's method is also correct..since this fn is continuous for x=2/3, we can check directly by finding its derivative and then checking from left side and right side..both methods are fine!!

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Best answer

We define the modulus function piecewise as :

F(x)   =   -(2 - 3x)   =   3x - 2   , for x < 2/3  and 

         =    2 - 3x    for x >= 2/3

So in order to check for differentiability , we check for left derivative and right derivative..

Here left derivative is w.r.t F(x)  =  3x - 2..

So d(F(x)) / dx = 0  ==> d/dx(3x - 2)   =   3 ..Hence left derivative is 3 at x = 3 [In fact at any point for x <= 2/3]

Now coming to right derivative , we have : F(x)  =   2 - 3x

So d(F(x)) / dx = 0  ==> d/dx(2 - 3x)   =   -3..Hence right derivative is -3 at x = 3 [In fact at any point x >= 2/3]

As at x = 2/3 , Left derivative != Right derivative , so we can conclude

F(x) is not differentiable at x = 2/3

answered by Veteran (64.9k points)  
edited by
we should find the left hand limit and right hand limit at that point to prove it is not differentiable
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