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I have a CST parameterization for a shape over an interval (0,1), so I have y as a function of x like so $$y = C(x)*s(x)$$ where $$C(x) = x^{n1}*(1-x)^{n2}$$ and $$S(x) = \sum_{i = 0}^{n} A_i(x)^i(1-x)^{n-i}$$ As such I clearly have the analytic expression of my shape. My question is how can I find the x coordinate for the maximum value of y on the surface on the interval (0,1) as a closed form expression. Is this possible? My first instinct was to use the 1st derivative being equal to 0, but that is true in more than one location obviously, and I need an analytic expression for the location of the maximum y for this rather gnarly function.

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