Consider the function f x = P x sin x - 2 , x ≠ 2 , and f x = 7 , x = 2 where P x is a polynomial such that P " x is always a constant and P 3 = 9 . If f x is continuous at x = 2 , then P 5 is equal to __________.

Step-by-step solution

We have, f x = P x sin x - 2 , 7 , x ≠ 2 x = 2 P " x = Constant ⇒ P x is a 2 degree polynomial Let P x = x - 2 a x + b f x is continuous at x = 2 f 2 + = f 2 - Now, lim x → 2 + P x sin x - 2 = 7 ⇒ lim x → 2 + x - 2 a x + b sin x - 2 = 7 ⇒ 2 a + b = 7 ⇒ P 3 = 3 - 2 3 a + b = 9 ⇒ 3 a + b = 9 ⇒ a = 2 , b = 3 Hence, P 5 = 5 - 2 2 . 5 + 3 = 3 . 13 = 39 .