calculus and analysis - Finding maximum or minimum of implicit functions

is there any built in function that can be used to find maximum or minimum of implicit functions?

For example, if we have the equation $$x^2 + y^2 = (2 x^2 + 2 y^2 - x)^2,$$ then we can visualize the set of all $(x,y)$ making the equation true using ContourPlot.

ContourPlot[
 x^2 + y^2 == (2 x^2 + 2 y^2 - x)^2, {x, -1, 2}, {y, -1, 1}, 
 AspectRatio -> Automatic]

Clearly, $y$ is not a function of $x$ but, in the neighborhood of most points on the graph, a function is implied, i.e. $y$ is implicitly a function of $x$. Is there any built in way to find the maximum and/or minimum value of this function (like what we have for the explicit functions)?

Answer

Maximize[{y, x^2 + y^2 == (2 x^2 + 2 y^2 - x)^2}, {x, y}]

{(3 Sqrt[3])/8, {x -> 3/8, y -> (3 Sqrt[3])/8}}