Let M and m be respectively the absolute maximum and the absolute minimum values of the function, f x = 2 x 3 - 9 x 2 + 12 x + 5 in the interval [ 0 , 3 ] . Then M - m is equal to

Step-by-step solution

Given f x = 2 x 3 - 9 x 2 + 12 x + 5 ,   x ∈ 0 , 3 Differentiating w.r.t x , ⇒ f ′ x = 6 x 2 - 18 x + 12 ⇒ f ′ x = 6 x - 1 x - 2 For maxima/minima f ' x = 0 ⇒ 6 x - 1 x - 2 = 0 ⇒ x = 1 , 2 Now, f " x = 12 x - 18 So at x = 1 ,   f " x < 0 and at x = 2 ,   f " x > 0 So, for absolute maxima/minima ⇒ f 1 = 10 ⇒ f 2 = 9 ⇒ f 0 = 5 ⇒ f 3 = 14 ⇒ M = 14 ⇒ m = 5 ∴  M - m = 9