I interpret your question as find the extrema of the function f[x,y] where x and y lie on some curve. In that case:
f[x_, y_] := 10 Log[x^2 + 2 y^2 + 1] - x^2 - y^2
Plot[f[Cos[t], Sin[t]], {t, 0, 2 Pi}]
Plot3D[f[x, y], {x, -2, 2}, {y, -2, 2}, RegionFunction -> ((#1^2 + #2^2 < 1) &)]
Solve[D[f[Cos[t], Sin[t]], t] == 0, t]
or in the g[t] form
g[t_] := f[{x[t], y[t]}
where {x[t],y[t]} is your curve.
