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Question

Mathematics Question on Functions

If f(x)=x3x2f(1)+xf(2)f(3),xRf(x)=x^3-x^2 f^{\prime}(1)+x f^{\prime \prime}(2)-f^{\prime \prime \prime}(3), x \in R, then

A

2f(0)f(1)+f(3)=f(2)2 f(0)-f(1)+f(3)=f(2)

B

f(3)f(2)=f(1)f(3)-f(2)=f(1)

C

3f(1)+f(2)=f(3)3 f(1)+f(2)=f(3)

D

f(1)+f(2)+f(3)=f(0)f(1)+f(2)+f(3)=f(0)

Answer

2f(0)f(1)+f(3)=f(2)2 f(0)-f(1)+f(3)=f(2)

Explanation

Solution

The correct answer is (A) : 2f(0)f(1)+f(3)=f(2)2 f(0)-f(1)+f(3)=f(2)
f(x)=x3−x2f′(1)+xf′′(2)−f′′′(3),x∈R
Let f′(1)=a,f′′(2)=b,f′′′(3)=c
f(x)=x3−ax2+bx−c
f′(x)=3x2−2ax+b
f′′(x)=6x−2a
f′′′(x)=6
c=6,a=3,b=6
f(x)=x3−3x2+6x−6
f(1)=−2,f(2)=2,f(3)=12,f(0)=−6
2f(0)−f(1)+f(3)=2=f(2)