Verified Solution • 2025–26 Session moderate

QJEEM23Ma0ad6: JEE Main 2023 Mathematics - Class 12 JEEM23Ma0ad6 Solution: If and , then the value of is equal to ____________.

If f(x)=x2+g′(1)x+g′′(2)f(x)=x^{2}+g^{\prime}(1) x+g^{\prime \prime}(2) and g(x)=f(1)x2+xf′(x)+f′′(x)g(x)=f(1) x^{2}+x f^{\prime}(x)+f^{\prime \prime}(x) , then the value of f(4)−g(4)f(4)-g(4) is equal to ____________.

Step-by-Step Verified Solution

Let g′(1)=ag^{\prime}(1)=a and g′′(2)=bg^{\prime \prime}(2)=b

⇒f(x)=x2+ax+b\Rightarrow f(x)=x^{2}+a x+b

Now, f(1)=1+a+b;f′(x)=2x+a;f′′(x)=2f(1)=1+a+b ; f^{\prime}(x)=2 x+a ; f^{\prime \prime}(x)=2

g(x)=(1+a+b)x2+x(2x+a)+2g(x)=(1+a+b) x^{2}+x(2 x+a)+2

⇒g(x)=(a+b+3)x2+ax+2\Rightarrow g(x)=(a+b+3) x^{2}+a x+2

⇒g′(x)=2x(a+b+3)+a⇒g′(1)=2(a+b+3)\Rightarrow g^{\prime}(x)=2 x(a+b+3)+a \Rightarrow g^{\prime}(1)=2(a+b+3) +a=a+a=a

⇒a+b+3=0\Rightarrow a+b+3=0 .......(1)

g′′(x)=2(a+b+3)=bg^{\prime \prime}(x)=2(a+b+3)=b

⇒2a+b+6=0\Rightarrow 2 a+b+6=0 ........(2)

Solving (i) and (ii), we get

a=−3a=-3 and b=0b=0

f(x)=x2−3xf(x)=x^{2}-3 x and g(x)=−3x+2g(x)=-3 x+2

f(4)=4f(4)=4 and g(4)=−12+2=−10g(4)=-12+2=-10

⇒f(4)−g(4)=16−2=14\Rightarrow f(4)-g(4)=16-2=14

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