Value of A and B, so that the function f(x) defined by f(x)=x+A2sin⁡x  if 0≤x<π4,2x+cot⁡x+B  if π4≤x<π2, becomes continuous are Acos⁡2x−Bsin⁡x if π2≤x≤π

Here we will check continuity at x=π/4 and x=π/2At          x=π/4.               limx→π−4fx=limx→π−4x+A2sinx=π4+Aand     limx→π+4  fx=limx→π+42xcotx+B=π2+BFor confinuity at x=π/4                π4+A=π2+B⇒A−B=π4Again, limx→π−2  fx=limx→π−22xcotx+B=Band,     limx→π+2  fx=limx→π+22Acos2x−Bsinx=−A−B,so f is continuous at this point if              −A−B=B⇒A=−2BSolving (i) and (ii), A=π/6 and B=−π/12Hence (1)is the correct answer.