Will you compare coefficients?

Calculus Level 5

C1:f(x)=x6+Ax5+Bx4+Cx3+Dx2+Ex+FC2:g(x)=Px+Q\begin{aligned} C_1: f\left( x \right) &=x^6+Ax^5+Bx^4+Cx^3+Dx^2+Ex+F \\ C_2: g\left( x \right) &=Px+Q \end{aligned}

Consider the curves C1C_1 and C2C_2 where A,B,C,D,E,F,P,QRA,B,C,D,E,F,P,Q \in \mathbb{R}.

Now it is given that f(x)f\left( x \right) touches g(x)g\left( x \right) at x=1,2,3x=1,2,3. Let A\mathcal{A} be the area bounded between the two curves C1C_1 and C2C_2.

If A\mathcal{A} can be expressed in the form mn\frac{m}{n}, find m+nm+n.

Details And Assumptions

  • m,nm,n are co-prime integers.

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