If the function f (x)= array cl a x- b x c x- b x & , if x 0 -1 & , if x=0 array . is continuous at x=0 , then a^2, b^2, c^2 are in

Step-by-step solution

Continuity at x = 0 requires: _ x 0 f(x) = f(0) = -1 Evaluating the limit: _ x 0 ax - bx cx - bx Apply L'Hôpital's Rule twice: _ x 0 -a ax + b bx -c cx + b bx = _ x 0 -a^2 ax + b^2 bx -c^2 cx + b^2 bx Substituting x = 0 : -a^2 1 + b^2 1 -c^2 1 + b^2 1 = b^2 - a^2 b^2 - c^2 Equate to f(0) = -1 : b^2 - a^2 b^2 - c^2 = -1 b^2 - a^2 = -(b^2 - c^2) b^2 - a^2 = -b^2 + c^2 2b^2 = a^2 + c^2 Thus a^2 , b^2 , and c^2 form an arithmetic progression with b^2 as the middle term. B