Suppose that $f$ is a twice differentiable function such that $$f(a) = 0,\, f(b) = 2,\, f(c) = − 1,\, f(d) = 2,\, f(e) = 0,$$ where $a < b < c < d < e$. What is the minimum number of zeroes of $g(x) = 2f'(x) + f''(x) f(x)$ in the interval $[a, e]$?
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