Calculating the MLE for mu(x) in a regression model Say we have the following regression model:
$$Y_i = \alpha + \beta(x_i - \mathrm{mean}(x)) + R_i$$
where $R_1,\ldots,R_{20} \sim G(0, \sigma)$
If we have $\mu(x) = \alpha + \beta(x - \mathrm{mean}(x))$, how do I go about finding the MLE of $\mu(5)$?
I have a given data set with some calculations done for me, but not sure how to approach this?

Michael Hardy is correct. Let l(θ) be the log of the likelihood function given the observed x and the parameter θ. Then you maximized the likelihood by finding θ such that ∂/∂θ l(θ) =0.  Now suppose we take the function g(θ) (in your case θ =(α, β) and g(α, β)=α+β(x−mean(x)). Consider l(g(θ)).  This is maximized by set partial derivatives with respect to α and β to 0.  For simplicity lets look at the one parameter case. ∂/∂θ l(g(θ))= ∂/∂θ l(g) ∂/∂θ g(θ) by the chain rule.  The loglikelihood is has partial derivative 0 for g(θ) at the same theta that the loglikeihood for theta has partial derivative =0.  Hence the maximum likelihood estimator for a differentiable function of theta is maximized at the function evaluated at the maximum likelihood estimator for theta.  
