The sum of all the local minimum values of the twice differentiable function f : R → R defined by f x = x 3 - 3 x 2 - 3 f " ( 2 ) 2 x + f " 1 is:

Step-by-step solution

f x = x 3 - 3 x 2 - 3 f " ( 2 ) 2 x + f " 1       … i ⇒ f ' x = 3 x 2 - 6 x - 3 2 f " ( 2 )       … ii ⇒ f " x = 6 x - 6       … iii Then, f " ( 2 ) = 12 - 6 = 6 f " ( 1 ) = 0 Now, f ' x = 3 x 2 - 6 x - 3 2 f " ( 2 ) ⇒ f ' x = 3 x 2 - 6 x - 3 2 × 6 ⇒ f ' ( x ) = 3 x 2 - 6 x - 9 For maxima/minima, we have f ' x = 0 ⇒ 3 x 2 - 6 x - 9 = 0 ⇒ x 2 - 2 x - 3 = 0 ⇒ x 2 - 3 x + x - 3 = 0