Let f ( x ) = x 3 + x 2 f ' ( 1 ) + x f " ( 2 ) + f ''' ( 3 ) , x ∈ R . Then f ' ( 10 ) is equal to

Step-by-step solution

Given: f ( x ) = x 3 + x 2 f ' ( 1 ) + x f " ( 2 ) + f ''' ( 3 ) ⇒ f ' ( x ) = 3 x 2 + 2 x f ' ( 1 ) + f " ( 2 ) . . . i ⇒ f " ( x ) = 6 x + 2 f ' ( 1 ) . . . i i ⇒ f ''' ( x ) = 6 ⇒ f ''' ( 3 ) = 6 . . . i i i Using i i , ⇒ f " ( 2 ) = 12 + 2 f ' ( 1 ) Putting this value in equation i , ⇒ f ' ( 1 ) = 3 + 2 f ' ( 1 ) + 12 + 2 f ' ( 1 ) ⇒ 0 = 15 + 3 f ' ( 1 ) ⇒ f ' ( 1 ) = - 5 . . . i v ⇒ f " ( 2 ) = 12 + 2 - 5 ⇒ f " ( 2 ) = 2 . . . v ⇒ f ( x ) = x 3 + x 2 · ( - 5 ) + x · ( 2 ) + 6 ⇒ f ' ( x ) = 3 x 2 - 10 x + 2 ⇒ f