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1)

Let  S=(tR:f(x)=∣xπ.ex1)sinx is not  differentiable at t). Then, the set S is equal to


A) ϕ (an empty set)

B) { 0}

C) {π}

D) {0,π}

Answer:

Option A

Explanation:

We have

  f(x)=∣xπ.(ex1)sinx

{(xπ)(ex1)sinx,x<0(xπ)(ex1)sinx,0x<π(xπ)(ex1)sinx,xπ

We check the differentiability at x=0 and  π

 We have,

{(xπ)(ex1)cosx+(ex1)sinx+(xπ)sinxex(1),x<0[(xπ)(ex1)cosx+(ex1)sinx+(xπ)sinxex],0<x<π(xπ)(ex1)cosx+(ex1)sinx+(xπ)sinxex,x>π

Clearly,      limx0f(x)=0=limx0+f(x)

and             limxπf(x)=0=limxπ+f(x)

   f is differentiable  at x=0 and x= \pi

  Hence, f is differentiable for all x