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Calculus Level 5

Define \( g_n (t) = e^{t + e^{t + e^{t + \cdots}}} \), \(n\) times. And \(\displaystyle f(t) =\lim_{n\to\infty} g_n (t) \). Then which of the following is/are true?

(a): \(f : \mathbb R^- \to \mathbb R^+ \).
(b): \(f(-1) = 1 \).
(c): \(f\) is one-to-one.
(d): \(f \) is increasing on \(\mathbb R\).
(e): \(f\) exists for all \(t \in \mathbb R\).
(f): \(f\) is increasing on \(\mathbb R^- \) excluding \((-1,0) \).

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