Question

If $f(x)=x^3-x^2 f^{\prime}(1)+x f^{\prime \prime}(2)-f^{\prime \prime \prime}(3), x \in R$, then

• A. $2 f(0)-f(1)+f(3)=f(2)$
• B. $f(3)-f(2)=f(1)$
• C. $3 f(1)+f(2)=f(3)$
• D. $f(1)+f(2)+f(3)=f(0)$

Answer 1

Answer: (A)

$f(x)=x^3-x^2 f^{\prime}(1)+x f^{\prime \prime}(2)-f^{\prime \prime \prime}(3), x \in R$
Let $f ^{\prime}(1)= a , f ^{\prime \prime}(2)= b , f ^{\prime \prime \prime}(3)= c$
$f ( x )=x^3- a x ^2+ b x - c $
$ f ^{\prime}(x)=3 x^2-2 a x+b$
$ f^{\prime \prime}(x)=6 x-2 a$
$ f^{\prime \prime \prime}(x)=6$
$c=6, a=3, b=6$
$ f(x)=x^3-3 x^2+6 x-6$
$ f(1)=-2, f(2)=2, f(3)=12, f(0)=-6$
$2 f(0)-f(1)+f(3)=2=f(2)$