Let a function g : 0 , 4 → R be defined as g x = max t 3 - 6 t 2 + 9 t - 3 0 ≤ t ≤ x , 0 ≤ x ≤ 3 4 - x , 3 < x ≤ 4 then the number of points in the interval 0 , 4 where g ( x ) is NOT differentiable, is _________.

Step-by-step solution

We have, g x = max   t 3 - 6 t 2 + 9 t - 3 0 ≤ t ≤ x , 0 ≤ x ≤ 3 4 - x , 3 < x ≤ 4 Let f x = x 3 - 6 x 2 + 9 x - 3 ⇒ f ' ( x ) = 3 x 2 - 12 x + 9 ⇒ f ' ( x ) = 3 ( x - 1 ) ( x - 3 ) For critical points: f ' x = 0 ⇒ x = 1 ,   3 And, f " x = 6 x - 12 Then, f " 1 = 6 - 12 = - 6 < 0 (maxima) f " 3 = 6 > 0 (minima) Hence, g x = f ( x ) , 0 ≤ x ≤ 1 1 , 1 ≤ x ≤ 3 4 - x , 3 < x ≤ 4 Hence, g x is c