Prove that any function $f:\mathbb{R}\to \mathbb{R}$ satisfying $|f(x)-f(y)|\leq c(x-y)^2$ for all $x, y$ and some $c\geq 0$ must be constant.
Let $f$ be twice-differentiable on $(0,\infty)$ and $f''(x)$ be bounded. Prove that $$ \lim_{x\to\infty} f'(x)=0 $$ if $\lim_{x\to\infty} f(x)=0$.
Prove that for any $n\in \mathbb{N}$ the polynomial function $$ p_n(x)=1+x+\frac{x^2}{2! }+\dots+\frac{x^n}{n!} $$ can have only simple real roots, that is roots of multiplicity 1.
Prove or disprove: there is a function $f$ on $\mathbb{R}^n$ with $n>1$ so that all first order partial derivaties $\frac{\partial f}{\partial x_j}$ exist everywhere and that $f$ is continuous but not differentiable at $x=0$.
Let $f, g: \mathbb{R}^n\to \mathbb{R}^m $ be differentiable maps and show that the product map $$ g\cdot f=g^\top f:\mathbb{R}^n \to \mathbb{R}\, ,\: x\mapsto g(x)\cdot f(x)=g(x)^\top f(x) $$ is differentiable and that $$ D\bigl( g^\top f\bigr)(x)=g(x)^\top Df(x) +f(x)^\top Dg(x)\, ,\: x\in \mathbb{R}^n. $$ Notice that $Dg^\top (x)=Dg(x)^\top$ for $x\in \mathbb{R}^n$.
Consider the polynomial function $$ f(x)=x^3+ax^2+bx+c\, ,\: x\in \mathbb{R}\, , $$ with real coefficients $a$, $b$ and $c$. Observe that when $a=0$, $b=-1$ and $c=0$ the equation $f(x)=0$ has three distinct real solutions, namely $u=1$, $v=-1$ and $w=0$. Use the Implicit Function Theorem to show that the solutions $u$, $v$, $w$ of $f(x)=0$ can be expressed as continuously differentiable functions of the coefficients $a$, $b$, $c$, when $(a,b,c)$ is close to $(0,-1,0)$.
Consider the matrix-valued function $$ f:\mathbb{R}^{n\times n}\to\mathbb{R}^{n\times n}\, ,\: M\mapsto M^3. $$ Is this function differentiable and, if yes, what is its derivative? Justify your answer.
Find the minimizers of $f$ given by $$ f(x_1, x_2, x_3)= x_1^2+3x_2^2 -x_2+3x_3 $$ over the set $\big\{ x\in \mathbb{R}^3\, \big |\, 2 x_3=x_1^2+x_2^2\big\}$.
Given $n$ distinct points $(x_1, y_1),\dots, (x_n, y_n)$ in $\mathbb{R}^2$, find the equation of a line $y=ax+b$ such that $$ \sum_{j=1}^n (a x_j+b-y_j)^2 $$ is minimized. Such line is called linear regression of the given $n$ points.
Determine the maximum value of the function $f:\mathbb{R}^n\to \mathbb{R}$ given by the formula $$ f(x)=\sum_{j=1}^n x_j\, ,\: x\in \mathbb{R}^n, $$ on the unit sphere. Explain why the value you obtain is really the desired maximum.