Extrapolation

Calculus Level 5

Observe the graph above. The graphed functions \(f(x), g(x)\) are both polynomials in \(x\) with \(f(x)\) being a degree 10 polynomial.

It is given that \(f(x)\) is monotonically decreasing in the interval \((-\infty, \alpha)\) and monotonically increasing in the interval \((\beta, \infty)\); \(g(x)\) is monotonically increasing in the interval \((-\infty, \alpha)\) and monotonically decreasing in the interval \((\beta, \infty)\).

With this information, if \(x_{f, m}, x_{g, n}\) are the (not necessarily distinct, possibly complex) roots of \(f(x), g(x)\) respectively, find the maximum possible value of

\[\deg(f) \cdot \deg(g) \cdot \text{sgn} \left (\prod_{m=1}^{\deg(f)} x_{f, m} \right ) \cdot \text{sgn} \left (\prod_{n=1}^{\deg(f)} x_{g, n} \right )\]

where \(\deg(f)\) is the degree of \(f\) and \(\text{sgn}(x)\) is the sign function.

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