Let f : R → R be defined as f x = 2 sin - π x 2 , if x < - 1 a x 2 + x + b , if - 1 ≤ x ≤ 1 sin π x , if x > 1 If f x is continuous on R , then a + b equals :

Step-by-step solution

f x is continuous on R ⇒ f 1 - = f 1 = f 1 + a + 1 + b = lim x → 1 + sin πx a + 1 + b = 0 ⇒ a + b = - 1 ⇒ Also f - 1 - = f - 1 = f - 1 + lim x → - 1 2 sin - π x 2 = a - 1 + b a - 1 + b = 2 Either a - 1 + b = 2 or a - 1 + b = - 2 a + b = 3       . . . 2 or a + b = - 1       . . . 3 from 1 and 2 ⇒ a + b = 3 = - 1 (reject) from 1 and 3 ⇒ a + b = - 1