Will you compare coefficients?

Calculus Level 5

\[\begin{align} C_1: f\left( x \right) &=x^6+Ax^5+Bx^4+Cx^3+Dx^2+Ex+F \\ C_2: g\left( x \right) &=Px+Q \end{align}\]

Consider the curves \(C_1\) and \(C_2\) where \(A,B,C,D,E,F,P,Q \in \mathbb{R}\).

Now it is given that \(f\left( x \right)\) touches \(g\left( x \right)\) at \(x=1,2,3\). Let \(\mathcal{A}\) be the area bounded between the two curves \(C_1\) and \(C_2\).

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

Details And Assumptions

  • \(m,n\) are co-prime integers.

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