This shows that the function has only one stationary point, which is a local minimum, with a value of Sqrt[2]:
f = y^2/4 + Sqrt[(x - y^2/4)^2 + (E^x - y)^2] + 1;
grad = D[f, {{x, y}}];
stationary = Solve[grad == 0, {x, y}, Reals]
f /. stationary[[1]] // FullSimplify
hess = D[grad, {{x, y}}] /. stationary[[1]] // FullSimplify;
Eigenvalues[hess] // N
