What hypothesis of the picard-lindel of theorem is violate

Question

Lipschitz Continuity and Initial Value Problems for ODE's

Explain the steps on how to solve the problem below:

1. Find a Lipschitz constant, K, for the function f(u, t) = u³ + tu² which shows that f is Lipschitz in u on the set 0 ≤ u ≤ 2, 0 ≤ t ≤ 1.

2. Show that the function f(u, t) = t√u, is not Lipschitz in u on [0, 1] × [0, 2].

3. Find two solutions to the initial value problem y′ = |y|1/2 , y(0) = 0. What hypothesis of the Picard-Lindel¨of Theorem is violated.