Q1. By computing appropriate Lipschitz constants, show that the following functions satisfy
Lipschitz condition on the sets D indicated:
(a) $g(t, u) = 4t^2 + u^2$, on $D: |t| \le 1, |u| \le 1$.
(b) $g(t, u) = t^2 \cos^2 u + u \sin^2 t$, on $D: |t| \le 1, |u| < \infty$
(c) $g(t, u) = t^2 e^{-tu^2}$, on $D: |t| \le 1, |u| < \infty$ and on $D: 0 \le t \le a, |u| < \infty$ (here $a > 0$ is a constant)
Q2. Show that the following functions do not satisfy the Lipschitz condition in the region indicated:
(a) $g(t, u) = \frac{\sin u}{t}$, $g(0, u) = 0$ on $D: |t| \le 1, |u| < \infty$.
(b) $g(t, u) = \frac{e^t}{u^2}$, $g(t, 0) = 0$, on $D: |t| \le 2, |u| \le \frac{1}{2}$
