For each choice of f (t, y) given in parts (a)-(d): i. Does f satisfy a Lipschitz

Question:

For each choice of f (t, y) given in parts (a)-(d):
i. Does f satisfy a Lipschitz condition on D = {(t, y) | 0 ≤ t ≤ 1, −∞ < y < ∞}?
ii. Can Theorem 5.6 be used to show that the initial-value problem
y' = f (t, y), 0≤ t ≤ 1, y(0) = 1,
is well-posed?
a. f (t, y) = et−y
b. f (t, y) = (1 + y)/(1 + t)
c. f (t, y) = cos(yt)
d. f (t, y) = y2/(1 + t)