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

The correct answer is 14.
f(x)=x2+g′(1)x+g′′(2)
f′(x)=2x+g′(1)
f′′(x)=2
g(x)=f(1)x2+x[2x+g′(1)]+2
g′(x)=2f(1)x+4x+g′(1)
g′′(x)=2f(1)+4
g′′(x)=0
2f(1)+4=0
f(1)=−2
−2=1+g′(1)=g′(1)=−3
So f′(x)=2x−3
f(x)=x2−3x+c
c=0
f(x)=x2−3x
g(x)=−3x+2
f(4)−g(4)=14