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Geometry Level 5

Let \[f(x)= 1+3x^2 + 5x^4 + 7x^6+ \cdots + 21x^{20} , \quad x \in \mathbb{R} \]

and \[g(x) = -x^2 + 4 \cos^2b- 4 \sin b -7, \quad b \in \mathbb{R} \]

If \(d\) is the shortest distance between \(f(x)\) and \(g(x)\) and \(d_1,d_2\) are the greatest and least values of \(d\) respectively, then \(\dfrac {d_1}{d_2}=?\)

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