Analytically minimizing a quadratic function

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How is the "analytically minimizing the quadratic function" done? Baffled...

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It is of course subjective, what is "analytical" and what isn't but from the context the author clearly means that we can write a closed form expression for $x(w)$ where the infimum over $x$, involved in the definition of $g(w)$, is attained. This is $x(w) = [A^TWA]^{-1}A^TWb$. The last expression for $g(w)$ is obtained by replacing $x$ with $x(w)$.