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Question

Let f(x)=5x+1,              x2x0(5+|1t|) dt,  x>2
then which of the following statement(s) is /are incorrect?
  1. f(x) is continuous but not differentiable at x=2
  2. f(x) is not continuous at x=2
  3. f(x) is differentiable for all xR
  4. f(x) is differentiable for x>2


Solution

The correct options are
B f(x) is not continuous at x=2
C f(x) is differentiable for all xR
For x>2,
f(x)=x0(5+|1t|) dt        =10(5+|1t|) dt+x1(5+|1t|) dt
       =10(6t) dt+x1(4+t) dt
       =[6tt22]10+[4t+t22]x1
       =612+4x+x22412
       =x22+4x+1

f(x)=5x+1,             x2x22+4x+1,   x>2

At x=2,  f(2)=11
limx2f(x)=11
limx2+f(x)=2+8+1=11
So f(x) is continuous at x=2.

f(x)={5,              x2x+4,       x>2

At x=2
limx2f(x)=5
limx2+f(x)=2+4=6
limx2f(x)limx2+f(x)
So f(x) is not differentiable at x=2
Therefore f(x) is differentiable for all xR{2}

 

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