KVPY Sample Paper KVPY Stream-SX Model Paper-7

Let \[f\,:\,R\to R\] be a function such that \[f\,(x)={{x}^{3}}+{{x}^{2}}f'(1)+xf''(2)+f''(3),\]\[x\,\in R.\] Then f(2) equals:

A) \[-4\]

B) 30

C) \[-2\]

D) 8

Correct Answer: C

Solution :

\[f(x)={{x}^{3}}+a{{x}^{2}}+bx+c\]

\[f'(x)=3{{x}^{2}}+2ax+b\]

\[f''(x)=6x+2a\]

\[f'''(x)=6a=f'(1)=3+2a+b\]

\[\Rightarrow \] \[a+b=-3\]

\[b=f''=12+2a\]

\[\Rightarrow \] \[2a-b=-12\]

\[c=f'''(3)\]

\[\Rightarrow \] \[c=6\]

and \[a=-5,b=2\]

\[\Rightarrow \] \[f(x)={{x}^{2}}-5{{x}^{2}}+2x+6\]

\[\Rightarrow \] \[f(2)=8-20+4+6=-2\]