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$p(0)\geq c'(0)$ is assumed so that the inverse demand curve originates above the marginal cost curve.

But what is the intuition behind this assumption? The highest willing-to-pay consumer is more than the marginal cost of producing the first unit? Is this going to be the variable cost of producing the first unit?

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You intuition about the assumption is correct.

Given the usual assumptions of downward sloping demand curve ($p'(q)<0$) and non-decreasing supply curve ($c''(q)\ge0$), if $p(0)<c'(0)$, then demand would lie entirely below supply. This means the equilibrium quantity must be zero---as if the market did not exist. Therefore, for the analysis to be non-trivial, the assumption of $p(0)\ge c'(0)$ is maintained.

With the use of calculus, we are assuming implicitly that quantities are infinitely divisible. Thus $c'(0)$ can be roughly interpreted as the cost of producing the "first infinitesimal unit" of the good. Mathematically, \begin{equation} c'(0)=\frac{\mathrm d\, TC(q)}{\mathrm d\,q}\Bigg\vert_{q=0}=\frac{\mathrm d\,(FC+VC(q))}{\mathrm d\,q}\Bigg\vert_{q=0}=\frac{\mathrm d\,VC(q)}{\mathrm d\,q}\Bigg\vert_{q=0}. \end{equation} Thus $c'(0)$ is the derivative of the variable cost function, evaluated at $q=0$.

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