Minimum of f(x,y) = (1-x)^2 + 100*(y-x^2)^2
With 718 function evaluations using the downhill simplex method:
-> [x0,y0] = [1.0000000080322589, 1.0000000082460916]
-> f(x0,y0) = 6.177295895412975e-15
With 14 function evaluations using root finding:
-> [x0,y0] = [1.0000000005184788, 1.0000000010356465]
-> f(x0,y0) = 2.689921996230038e-19

Minima of f(x,y) = (x^2+y-11)^2 + (x+y^2-7)^2
With 1662 function evaluations using the downhill simplex method:
-> [x0,y0] = [2.9999997570895043, 1.9999999885633777] ∨ [-3.7793101023019244, -3.2831859065459583] ∨ [-2.8051178793478173, 3.131312463446214] ∨ [3.584428327902854, -1.8481265562889515]
-> f(x0,y0) = 2.240988704861553e-12 ∨ 1.2819558878469554e-12 ∨ 1.5056129063900136e-12 ∨ 2.3231424989158036e-14
With 240 function evaluations using root finding:
-> [x0,y0] = [2.999999999946565, 2.000000000182275] ∨ [-3.7793102536240024, -3.2831859913413934] ∨ [-2.8051180953122214, 3.1313125203977887] ∨ [3.5844283404115234, -1.848126527699191]
-> f(x0,y0) = 4.756604368865051e-19 ∨ 3.275641396697483e-18 ∨ 2.431380754382948e-15 ∨ 7.846844378460531e-18

tan(x) = x
With 134 function evaluations using the downhill simplex method:
-> x = 4.493409156799316
With 55 function evaluations using root finding:
-> x = -0.014368595286374707

exp(x*y)^z = x-y^z (NOTE: 1 eq., 3 unknowns!)
With 908 function evaluations using the downhill simplex method:
-> [x,y,z] = [0.1245717019428491, -0.992952447485831, -0.9999999999610923]