Let S = {(λ, μ) ∈ R × R : f(t) = (|λ|e|t| − μ)sin(2|t|), t ∈ R, is a differentiable function}. Then S is a subset of?

S = {(λ, μ) ∈ R × R : f(t) = (|λ|e|t| − μ)sin(2|t|), t ∈ R}
f(t) = (|λ|e|t| − μ)sin(2|t|) = (|λ|et − μ)sin(2t) t > 0
(|λ|e−t − μ)(−sin(2t)) t < 0
f′(t) = (|λ|et)sin(2t) + (|λ|et − μ)(2cos(t)) t > 0

• |λ|e−tsin(2t) + (|λ|e−t − μ)(−cos(2t)) t < 0
  Given f(t) is differentiable ∴ LHD = RHD at t = 0
  |λ| ⋅ sin(2(0)) + (|λ|e⁰ − μ)2cos(0) = |λ|e⁰sin(2(0)) + (|λ|e⁰ − μ)2cos(0)
  0 + (|λ| − μ)2 = 0
  (|λ| − μ)2 = 0
  |λ| = μ
  S = {(λ, μ) = λ ∈ R, μ ∈ [0, ∞)}
  Set S is subset of R × [0, ∞)