Given, f(x)=a0x2+a1x+a2 f′(0)=1 f′(1)=0 a0,a1,a2 are in A. G. P Common difference of AP=1 Common ratio of GP=2 A.P terms = a, a +1,a+2 G.P terms =y,ry,r2y ∴ AGP terms = ay, (a+1)ry,(a+2)r2y
∴a0=ay a1=(a+1)ry=(a+1)2y a2=(a+2)r2y=(a+2)4y Now, f′(x)=2xa0+a1 ∴f′(0)=a1=1 and f′(1)=2a0+a1=0 ⇒2a0+1=0 ⇒a0=−1∕2 ∴ay=−1∕2 and (a+1)2y=1 ⇒2ay+2y=1