The set of values of b for which local extremum values of the function f ( x ) are positive where f x = 2 3 a 2 x 3 - 5 a 2 x 2 + 3 x + b and maxima occurs at x = 1 3 is

Step-by-step solution

f ′ x = 2 a 2 x 2 - 5 a x + 3 = a x - 1 2 a x - 3 = 0 x = 1 a , 3 2 a If a > 0 , then local maxima occurs at x = 1 a and minima at x = 3 2 a ∵ maxima occurs at x = 1 a = 1 3 ⇒ a = 3 Minima occurs at x = 3 2 a = 1 2 f 1 2 > 0 ⇒ 3 8 + b > 0 ⇒ b > - 3 8 If a < 0 , then maxima shall occur at x = 3 2 a and minima at x = 1 2 a i.e. 3 2 a = 1 3 ⇒ a = 9 2 > 0 (not acceptable) Hence, b > - 3 8