Question

Problem 11. True or false? If true, justify. If false, give a counter example. a) If $f: \mathbb{R}^2 \to \mathbb{R}$ is continuous and $t_0, y_0 \in \mathbb{R}$, then the differential equation problem $f'(t) = f(t, y(t)), y(t_0) = y_0$ has a unique solution. b) If $f: \mathbb{R}^2 \to \mathbb{R}$ is a continuous function such that the partial derivatives $D_1f$ and $D_2f$ exist everywhere on $\mathbb{R}^2$, then $f$ is differentiable on $\mathbb{R}^2$. c) If $A: \mathbb{R}^n \to \mathbb{R}^m$ and $B: \mathbb{R}^m \to \mathbb{R}^k$ are linear transformations and $||A||$ is the operator norm, then $||BA|| \le ||B|| ||A||$. Problem 12. a) State and prove the mean value inequality for differentiable functions $f: E \to$ $\mathbb{R}^m$ defined on a convex set $E \subset \mathbb{R}^n$. [You may assume the mean value inequality for functions $f: \mathbb{R} \to \mathbb{R}^m$ without proof.] b) Give an example of a non-convex set $E$ and a differentiable function $f$ on $E$ for which the mean value inequality is violated.

          Problem 11. True or false? If true, justify. If false, give a counter example.
a) If $f: \mathbb{R}^2 \to \mathbb{R}$ is continuous and $t_0, y_0 \in \mathbb{R}$, then the differential equation problem
$f'(t) = f(t, y(t)), y(t_0) = y_0$
has a unique solution.
b) If $f: \mathbb{R}^2 \to \mathbb{R}$ is a continuous function such that the partial derivatives $D_1f$ and $D_2f$ exist
everywhere on $\mathbb{R}^2$, then $f$ is differentiable on $\mathbb{R}^2$.
c) If $A: \mathbb{R}^n \to \mathbb{R}^m$ and $B: \mathbb{R}^m \to \mathbb{R}^k$ are linear transformations and $||A||$ is the operator norm,
then
$||BA|| \le ||B|| ||A||$.
Problem 12.
a) State and prove the mean value inequality for differentiable functions $f: E \to$
$\mathbb{R}^m$ defined on a convex set $E \subset \mathbb{R}^n$. [You may assume the mean value inequality for
functions $f: \mathbb{R} \to \mathbb{R}^m$ without proof.]
b) Give an example of a non-convex set $E$ and a differentiable function $f$ on $E$ for which the
mean value inequality is violated.

Problem 11. True or false? If true, justify. If false, give a counter example.
a) If f: ℝ^2 →ℝ is continuous and t0, y0 ∈ℝ, then the differential equation problem
f'(t) = f(t, y(t)), y(t0) = y0
has a unique solution.
b) If f: ℝ^2 →ℝ is a continuous function such that the partial derivatives D1f and D2f exist
everywhere on ℝ^2, then f is differentiable on ℝ^2.
c) If A: ℝ^n →ℝ^m and B: ℝ^m →ℝ^k are linear transformations and ||A|| is the operator norm,
then
||BA|| ≤ ||B|| ||A||.
Problem 12.
a) State and prove the mean value inequality for differentiable functions f: E →
ℝ^m defined on a convex set E ⊂ℝ^n. [You may assume the mean value inequality for
functions f: ℝ→ℝ^m without proof.]
b) Give an example of a non-convex set E and a differentiable function f on E for which the
mean value inequality is violated.

Added by Remedios A.

Question

Please give Ace some feedback