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Generating a random monotonically increasing polynomial?

Given a polynomial $y : \mathbb{R} \mapsto \mathbb{R}$ of degree $p$: $$ y(x) = \sum_{k=0}^p c_k\, x^k,$$ can a random set of coefficients $\{c_0, \cdots ,c_p\}$ be generated such that $y$ is ...
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optimization of polynomial function with linear constraints

I have a polynomial function, which is actually the Cobb-Douglas production function, of the form $f(x,y) = \frac{\{x^{\alpha} y^{1-\alpha}\}^{1-\gamma}}{1-\gamma}$ with linear constraint K(x,y)= ...