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  4. If f(x)=x2+g prime(1) x+g prime prime(2) and g(x)=f(1) x2+x f prime(x)+f prime prime(x), then the value of f(4)-g(4) is equal to

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Q. 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 ______

JEE MainJEE Main 2023Limits and Derivatives

Solution:

$ 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\left[2 x+g^{\prime}(1)\right]+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$