If the functions f x = x 3 3 + 2 b x + a x 2 2 and g x = x 3 3 + a x + b x 2 , a ≠ 2 b have a common extreme point, then a + 2 b + 7 is equal to

Step-by-step solution

Given, f x = x 3 3 + 2 b x + a x 2 2 and g x = x 3 3 + a x + b x 2 Now differentiating the above function we get, f ' x = x 2 + 2 b + a x g ' x = x 2 + a + 2 b x Now given, f ' ( x ) = 0   a n d   g ' ( x ) = 0 have a common extreme point, So, x 2 + 2 b + a x = x 2 + a + 2 b x ⇒ 2 b - a - x 2 b - a = 0 ∴       x = 1 is the common extreme point Put x = 1 in f ' x = 0 or g ' x = 0 we get, 1 + 2 b + a = 0 ⇒ a + 2 b + 7 = 6