Differentiation Question 1
Question 1 - 24 January - Shift 2
If $f(x)=x^{3}-x^{2} f^{\prime}(1)+x f^{\prime \prime}(2)-f^{\prime \prime \prime}(3), x \in R$, then
(1) $3 f(1)+f(2)=f(3)$
(2) $f(3)-f(2)=f(1)$
(3) $2 f(0)-f(1)+f(3)=f(2)$
(4) $f(1)+f(2)+f(3)=f(0)$
Show Answer
Answer: (3)
Solution:
Formula: differentiation of some elementary functions
$f(x)=x^{3}-x^{2} f^{\prime}(1)+x f^{\prime \prime}(2)-f^{\prime \prime \prime}(3), x \in R$
Let $f^{\prime}(1)=a, f^{\prime \prime}(2)=b, f^{\prime \prime \prime}(3)=c$
$\mathbf{f}(\mathbf{x})=\mathbf{x}^{3}-\mathbf{a x ^ { 2 }}+\mathbf{b x}-\mathbf{c}$
$f^{\prime}(x)=3 x^{2}-2 ax+b$
$f^{\prime \prime}(x)=6 x-2 a$
$f^{\prime \prime \prime}(x)=6$
$c=6, a=3, b=6$
$f(x)=x^{3}-3 x^{2}+6 x-6$
$f(1)=-2, f(2)=2, f(3)=12, f(0)=-6$
$2 f(0)-f(1)+f(3)=2=f(2)$
