One of the great things about Mathematica's symbolic capabilities is that it can be used to illustrate mathematical definitions. This illustrates finding the minimum of x + y subject to x^2 + y^2 == 1, using the definition of constrained minimization.
In[2]:= Reduce[ForAll[{x, y}, x^2 + y^2 == 1, min <= x + y], min]
Out[2]= min <= -Sqrt[2]
In[3]:= Reduce[{x^2 + y^2 == 1, x + y == -Sqrt[2]}, {x, y}]
Out[3]= x == -(1/Sqrt[2]) && y == 1/Sqrt[2] - Sqrt[2]
