# 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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