Question

The third derivative of a function $f\left(x\right)$ vanishes for all $x$. lf $f\left(0\right)=1,f^{{}^{\prime }}\left(1\right)=2$, and $f^{{}^{\prime }}\left(2\right)=$ -1 then $f\left(x\right)=$

• A. $x^3+3x+2$
• B. $\frac{-x^2-3x+1}{2}$
• C. $\frac{x^2+3x-1}{2}$
• D. $\frac{-3x^2+10x+2}{2}$

Answer 1

The correct option is D $\frac{-3x^2+10x+2}{2}$

$Letf\left(x\right)=a{x}^2+bx+c\left(as{f}^{‘‘‘}\right(x)=0)$
$f\left(0\right)=c,c=1.....\left(1\right)$
$f^{\prime }\left(x\right)=2ax+b$
$f^{\prime \prime }\left(1\right)=2a+b=2$
$f^{\prime \prime }\left(2\right)=4a+b=-1$

$a=\frac{-3}{2},b=5,c=1$

$f\left(x\right)=\frac{3x^2+10x+2}{2}$