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Question

If fx=ax2-b,if x<11x     ,if x1 is differentiable at x = 1, find a, b.


Solution

Given: f(x) =ax2+b,     x<11x,           x1
 f(x) = -1x ,                          x<-1ax2-b,                     -1<x<11x,                               x1  

It is given that the given function is differentiable at x = 1.

We know every differentiable function is continuous. Therefore it is continuous at x=1. Then,

   lim  x1- f(x) = limx1+ f(x) limx1 ax2-b = limx1 1x a-b = 1                                       ...(i)


It is also differentiable at x=1. Therefore,

 (LHD at x = 1) = (RHD at x = 1)

limx1-f(x) - f(1)x-1 = limx1+f(x) - f(1)x-1 limx1 ax2-b - 1x-1= limx1 1x - 1x-1  limx1 ax2+1-a-1x-1=limx1 -(x-1)x-1                 Using (i)  limx1 a (x+1)= limx1 -1   2a=-1  a =-12
From (i), we have:
          a-b = 1-12 - b = 1 b =- 32

Hence, when a=-12 and b=-32 the function is differentiable at x = 1. 

Mathematics
RD Sharma XII Vol 1 (2015)
Standard XII

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