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1)

Let a,bϵR and f:R→ R  be defined by  f(x)=acos(|x3x|)+b|x|sin(|x3+x|). Then f is 


A) differentiable at x=0, if a=0 and b=1

B) differentiable at x=1, if a=1 and b=0

C) not differentiable at x=0, if a=1 and b=0

D) not differentiable at x=1, if a=1 and b=1

Answer:

Option A,B

Explanation:

f(x)=acos(|x3x|)+b|x|sin(|x3+x|)

   if  x3x0

cos|x3x|=cos(x3x)

            x3x0

cos|x3x|=cos(x3x)

cos(|x3x|)=cos(x3x),xϵR   .........(i)

Again , if   x3+x0

    |x|sin(|x3+x|)=xsin(x3+x)

x3+x0

 |x|sin(|x3+x|)=xsin((x3+x))

                 |x|sin(|x3+x|)=xsin(x3+x),xϵR  .......(ii)

f(x)=acos(|x3x|)+b|x|sin(|x3+x|)

  f(x)=acos(x3x)+bxsin(x3+x)     ........(iii)

which is clearly sum and composition of differential functions.

    Hence, f(x) is always continuous  and differentiable