Points of Maxima

Calculus Level 4

Let \(f(x)\) be a function defined on \(\mathbb R\) (the set of all real numbers) such that \[f'(x)=2010 (x-2009)(x-2010)^2(x-2011)^3(x-2014)^4\] for all \(x\in\mathbb R\).

If \(g(x)\) is a function defined on \(\mathbb R\) with values in the interval \((0,\infty)\) such that \(f(x)=\ln (g(x))\) for all \(x\in \mathbb R\), then find the number of points in \(\mathbb R\) at which \(g(x)\) has a local maximum.

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