Actually, TR solvers are not derivative free methods, users objective function F(x) = y f(x), where F(x): Rn ->Rm, m>=nshould bea twice differentiable function in Rn.
As of your function, it is unbounded, if t nears 0 and t < 0. Something likes that
F(x) = 1.935444e+44, t = -1.000000e-02
F(x) = 1.295015e+49, t = -9.000000e-03
F(x) = 1.393615e+55, t = -8.000000e-03
F(x) = 7.932340e+62, t = -7.000000e-03
F(x) = 1.736785e+73, t = -6.000000e-03
F(x) = 5.202701e+87, t = -5.000000e-03
F(x) = 2.697447e+109, t = -4.000000e-03
F(x) = 4.189477e+145, t = -3.000000e-03
F(x) = 1.010586e+218, t = -2.000000e-03
F(x) = inf, t = -1.000000e-03
F(x) = 1.200000e+00, t = 0.000000e+00
F(x) = 1.200000e+00, t = 1.000000e-03
F(x) = 1.200000e+00, t = 2.000000e-03
Did try to use solver for problems with boundary constraints [ T > 0] and use initial point [ T > 0 ]?