It is known that the function f (x) = ln (ex + a) (a is a constant) is an odd function on the real number set R, and the function g (x) = λ f (x) + SiNx (λ≤ - 1) is a decreasing function on the interval [- 1,1]; (1) find the value of a; (2) if G (x) ≤ T2 - λ T + 1 is constant on X ∈ [- 1,1], find the value range of T

It is known that the function f (x) = ln (ex + a) (a is a constant) is an odd function on the real number set R, and the function g (x) = λ f (x) + SiNx (λ≤ - 1) is a decreasing function on the interval [- 1,1]; (1) find the value of a; (2) if G (x) ≤ T2 - λ T + 1 is constant on X ∈ [- 1,1], find the value range of T

(1) If f (x) = ln (ex + a) is an odd function, then ln (ex + a) = - ln (E-X + a) is constant (2 points); (ex + a) (E-X + a) = 11 + ae-x + AEX + A2 = 1 ∧ a (ex + E-X + a) = 0 ∧ a = 0 (4 points) (2) and ∧ g (x) max = g (- 1) = - λ - sin1 (6 points) ∧ only - λ - sin1 ≤ T2 - λ T + 1, (8 points) ∧ (1-T) λ + T2 + sin1 + 1 ≥ 0 (where λ≤ - 1) Let H (λ) = (1-T) λ + T2 + sin1 + 1 (λ ≤ - 1), then 1 − t ≤ 0t − 1 + T2 + sin1 + 1 ≥ 0 (11 points) ‖ t ≥ 1t2 + T + sin1 ≥ 0 and T2 + T + sin1 ≥ 0, then 1 − t ≤ 1 (13 points)