### 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, $f^{\prime}(x)=2 x-3$

$f(x)=x^{2}-3 x+c$

$c=0$

$f(x)=x^{2}-3 x$

$g(x)=-3 x+2$

$f(4)-g(4)=14$