Question Number 10118 by Gaurav3651 last updated on 25/Jan/17

Let f:R→R be a function such that  f(x)=x^3 +x^2 f′(1)+xf′′(2)+f′′′(3)  for x∈R.  1)What is f(1) equal to?  2)What is f′(1) equal to?  3)What is f′′′(10) equal to?  For this question consider the following:  1) f(2)=f(1)−f(0)  2)f′′(2)−2f′(1)=12  which is/are correct?

Commented bynume1114 last updated on 25/Jan/17

Answer  1)f(1)=4  2)f′(1)=−5  3)f′′′(10)=6    1) f(2)=f(1)−f(0)  2)f′′(2)−2f′(1)=12  Both of these are correct.

Answered by nume1114 last updated on 25/Jan/17

f(x)=x^3 +x^2 f′(1)+xf′′(2)+f′′′(3) ...(A)  f′(x)=3x^2 +2xf′(1)+f′′(2) ...(B)  f′′(x)=6x+2f′(1) ...(C)  f′′′(x)=6 ...(D)  (D)⇒f′′′(3)=6 ...(E)          ⇒f′′′(10)=6  (B)⇒f′(1)=3+2f′(1)+f′′(2)          ⇒f′(1)+f′′(2)=−3 ...(F)  (C)⇒f′′(2)=12+2f′(1) ...(G)          ⇒f′′(2)−2f′(1)=^! 12  (F),(G)⇒f′(1)+[12+2f′(1)]=−3                   ⇒f′(1)=−5 ...(H)                   ⇒f′′(2)=2 ...(I)  (A),(E),(H),(I)  ⇒f(x)=x^3 −5x^2 +2x+6  ⇒f(1)=4  ⇒f(2)=−2  ⇒f(0)=6  ⇒f(2)=^! f(1)−f(0)