Problem Set 3

Let $(f_n)$ be real-valued, twice continuously differentiable on $[0,1]$, with $f_n(0) = f_n'(0) = 0$ and $\sup_{[0,1]} |f_n''| \leq 2$. Prove some subsequence converges uniformly.

Show each is a $G_\delta$ set:

(a) $(0, 2]$.

(b) The set $\mathcal{L}$ of Liouville numbers.

(c) $\{x : f'(x) = 0\}$ for a differentiable $f : \mathbb{R} \to \mathbb{R}$.

If $\vec F = (F_1, \dots, F_m): U \to \mathbb{R}^m$ and each $F_j$ is differentiable at $\vec a$, show $\vec F$ is differentiable at $\vec a$.

Under the hypotheses of the theorem (partials exist near $\vec a$ and continuous at $\vec a$), prove $f := F_j$ is differentiable at $\vec a$ with $Df(\vec a) = \nabla f(\vec a)^\top$.

Consider $C$ defined by

$\log\!\left(\cos(y + \tfrac{\pi}{12}) + \sin(y + \tfrac{\pi}{12})\right) - \log\!\left(\cos(y + \tfrac{\pi}{12}) - \sin(y + \tfrac{\pi}{12})\right) = 2x + \log(2 + \sqrt 3).$

Show that near $(0, \pi/12)$, $C$ is the graph of a $C^1$ function $y = g(x)$ satisfying $g'(x) = \cos(2g(x) + \pi/6)$, $g(0) = \pi/12$.

Let $f_1 = \tfrac{x}{y+z} + \tfrac{y}{x+z} + \tfrac{z}{x+y} - 4$, $f_2 = \tfrac{x^2}{y+z} + \tfrac{y^2}{x+z} + \tfrac{z^2}{x+y} - 42$. Let $C$ be where both vanish.

(a) Near $(11, 4, -1)$, represent $C$ as $(g_1(x), g_2(x))$.

(b) Find tangent line at $(11, 4, -1)$.