In[1]:= $Version
Out[1]= "10.0 for Microsoft Windows (64-bit) (September 9, 2014)"
In[2]:= FindMinimum[{x + y, x^2 + y^2 == 1}, {{x, 1}, {y, 1}}]
Out[2]= {1.41421, {x -> 0.707107, y -> 0.707107}}
I suppose this happens because the the starting point, maximum and minimum are on a straight line with the maximum nearer to the starting point.
In[7]:= FindMinimum[{x + y, x^2 + y^2 == 1}, {{x, 1}, {y, 2}}]
Out[7]= {-1.41421, {x -> -0.707107, y -> -0.707107}}