Application Of Derivatives Question 11
Question 11 - 01 February - Shift 1
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
Show Answer
Answer: (14)
Solution:
Formula: Successive differentiation
$f(x)=x^{2}+g^{\prime}(1) x+g^{\prime \prime}(2)$
$f^{\prime}(x)=2 x+g^{\prime}(1)$
$f^{\prime \prime}(x)=2$
$g(x)=f(1) x^{2}+x[2 x+g^{\prime}(1)]+2$
$g^{\prime}(x)=2 f(1) x+4 x+g^{\prime}(1)$
$g^{\prime \prime}(x)=2 f(1)+4$
$g^{\prime \prime}(x)=0$
$2 f(1)+4=0$
$f(1)=-2$
$-2=1+g^{\prime}(1)=g^{\prime}(1)=-3$
So, $f^{\prime}(x)=2 x-3$
$f(x)=x^{2}-3 x+c$
$c=0$
$f(x)=x^{2}-3 x$
$g(x)=-3 x+2$
$f(4)-g(4)=14$
