Question

If $f(x)=x^2+g^{\prime }(1)x+g^{\prime \prime }(2)$ and $g(x)=f(1)x^2+x{f}^{\prime }(x)+f^{\prime \prime }(x)$, then the value of $f(4)-g(4)$ is equal to ______

Answer 1

Answer: 14

$f(x)=x^2+g^{\prime }(1)x+g^{\prime \prime }(2)$
$f^{\prime }(x)=2x+g^{\prime }(1)$
$f^{\prime \prime }(x)=2$
$g(x)=f(1)x^2+x\left[2x+g^{\prime }(1)\right]+2$
$g^{\prime }(x)=2f(1)x+4x+g^{\prime }(1)$
$g^{\prime \prime }(x)=2f(1)+4$
$g^{\prime \prime }(x)=0$
$2f(1)+4=0$
$f(1)=-2$
$-2=1+g^{\prime }(1)=g^{\prime }(1)=-3$
So $f^{\prime }(x)=2x-3$
$f(x)=x^2-3x+c$
$c=0$
$f(x)=x^2-3x$
$g(x)=-3x+2$
$f(4)-g(4)=14$