Let $f$ be twice continuously differentiable on $(0,\infty)$ such that $\lim_{x \to 0^{+}}f'(x)=-\infty$ and $\lim_{x \to 0^{+}}f''(x)=\infty$. Show that $$\lim_{x\to 0^{+}}\frac{f(x)}{f'(x)}=0.$$

Find all differentiable functions $f:(0,\infty) \to \mathbb{R}$ such that $$f(b)-f(a)=(b-a)f’(\sqrt{ab}) \qquad \text{for all}\qquad a,b>0.$$ [i]Proposed by Orif Ibrogimov, National University of Uzbekistan[/i]

Let $f: \mathbb{R} \to \mathbb{R}$ be a differentiable function such that $$f(x)=f \left( \frac{x}{2} \right) + \frac{x}{2} f'(x), ~\forall x \in \mathbb{R}.$$ Prove that $f$ is a polynomial function of degree at most one. [hide=Note]The problem was posted quite a few times before: [url]https://artofproblemsolving.com/community/c7h100225p566080[/url] [url]https://artofproblemsolving.com/community/q11h564540p3300032[/url] [url]https://artofproblemsolving.com/community/c7h2605212p22490699[/url] [url]https://artofproblemsolving.com/community/c7h198927p1093788[/url] I'm reposting it just to have a more suitable statement for the [url=https://artofproblemsolving.com/community/c13_contests]Contest Collections[/url]. [/hide]

Let $f:[0,1] \rightarrow \mathbb{R}$ a non-decreasing function, $f \in C^1,$ for which $f(0) = 0.$ Let $g:[0,1] \rightarrow \mathbb{R}$ a function defined by \[ g(x) = f(x) + (x - 1) f'(x), \forall x \in [0,1]. \] a) Show that \[ \int_{0}^{1} g(x) \text{dx} = 0. \] b) Prove that for all functions $\phi :[0,1] \rightarrow [0,1],$ convex and differentiable with $\phi(0) = 0$ and $\phi(1) = 1,$ the inequality holds \[ \int_{0}^{1} g( \phi(t)) \text{dt} \leq 0. \]

Let $f,g:\mathbb R\to\mathbb R$ be continuous functions such that $g$ is differentiable. Assume that $(f(0)-g'(0))(g'(1)-f(1))>0$. Show that there exists a point $c\in (0,1)$ such that $f(c)=g'(c)$. [i]Proposed by Fereshteh Malek, K. N. Toosi University of Technology[/i]

Define $f(x)=\max \{\sin x, \cos x\} .$ Find at how many points in $(-2 \pi, 2 \pi), f(x)$ is not differentiable? [list=1] [*] 0 [*] 2 [*] 4 [*] $\infty$ [/list]

Let $f: [0,\infty) \to \mathbb{R}$ be a continuously differentiable function satisfying $$f(x) = \int_{x - 1}^xf(t)\mathrm{d}t$$ for all $x \geq 1$. Show that $f$ has bounded variation on $[1,\infty)$, i.e. $$\int_1^{\infty} |f'(x)|\mathrm{d}x < \infty.$$

Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function such that $\lim \limits_{x\to \infty}f'(x)=1$, then [list=1] [*] $f$ is increasing [*] $f$ is unbounded [*] $f'$ is bounded [*] All of these [/list]

Find all differentiable functions $f, g:[0,\infty) \to \mathbb{R}$ and the real constant $k\geq 0$ such that \begin{align*} f(x) &=k+ \int_0^x \frac{g(t)}{f(t)}dt \\ g(x) &= -k-\int_0^x f(t)g(t) dt \end{align*} and $f(0)=k, f'(0)=-k^2/3$ and also $f(x)\neq 0$ for all $x\geq 0$.\\ \\ [i] (Nora Gavrea)[/i]

We consider a function $f:\mathbb{R} \rightarrow \mathbb{R}$ for which there exist a differentiable function $g : \mathbb{R} \rightarrow \mathbb{R}$ and exist a sequence $(a_n)_{n \geq 1}$ of real positive numbers, convergent to $0,$ such that \[ g'(x) = \lim_{n \to \infty} \frac{f(x + a_n) - f(x)}{a_n}, \forall x \in \mathbb{R}. \] a) Give an example of such a function f that is not differentiable at any point $x \in \mathbb{R}.$ b) Show that if $f$ is continuous on $\mathbb{R}$, then $f$ is differentiable on $\mathbb{R}.$

$\textbf{(a)}$ Let $a,b \in \mathbb{R}$ and $f,g :\mathbb{R}\rightarrow \mathbb{R}$ be differentiable functions. Consider the function $$h(x)=\begin{vmatrix} a &b &x\\ f(a) &f(b) &f(x)\\ g(a) &g(b) &g(x)\\ \end{vmatrix}$$ Prove that $h$ is differentiable and find $h'$. \\ \\ $\textbf{(b)}$ Let $n\in \mathbb{N}$, $n\geq 3$, take $n-1$ pairwise distinct real numbers $a_1<a_2<\dots <a_{n-1}$ with sum $\sum_{i=1}^{n-1}a_i = 0$, and consider $n-1$ functions $f_1,f_2,...f_{n-1}:\mathbb{R}\rightarrow \mathbb{R}$, each of them $n-2$ times differentiable over $\mathbb{R}$. Prove that there exists $a\in (a_1,a_{n-1})$ and $\theta, \theta_1,...,\theta_{n-1}\in \mathbb{R}$, not all zero, such that $$\sum_{k=1}^{n-1} \theta_k a_k=\theta a$$ and, at the same time, $$\sum_{k=1}^{n-1}\theta_kf_i(a_k)=\theta f_i^{(n-2)}(a)$$ for all $i\in\{1,2...,n-1\} $. \\ \\ [i](Sergiu Novac)[/i]

Find all differentiable functions $f, g:[0,\infty) \to \mathbb{R}$ and the real constant $k\geq 0$ such that \begin{align*} f(x) &=k+ \int_0^x \frac{g(t)}{f(t)}dt \\ g(x) &= -k-\int_0^x f(t)g(t) dt \end{align*} and $f(0)=k, f'(0)=-k^2/3$ and also $f(x)\neq 0$ for all $x\geq 0$.\\ \\ [i] (Nora Gavrea)[/i]

Let $f(x)$ be a differentiable function defined on the interval $(0,1)$ such that $|f'(x)| \leq M$ for $0<x<1$ and a positive real number $M.$ Prove that $$\left| \int_{0}^{1} f(x)\; dx - \frac{1}{n} \sum_{k=1}^{n} f\left(\frac{k}{n} \right) \right | \leq \frac{M}{n}.$$

$\textbf{(a)}$ Let $a,b \in \mathbb{R}$ and $f,g :\mathbb{R}\rightarrow \mathbb{R}$ be differentiable functions. Consider the function $$h(x)=\begin{vmatrix} a &b &x\\ f(a) &f(b) &f(x)\\ g(a) &g(b) &g(x)\\ \end{vmatrix}$$ Prove that $h$ is differentiable and find $h'$. \\ \\ $\textbf{(b)}$ Let $n\in \mathbb{N}$, $n\geq 3$, take $n-1$ pairwise distinct real numbers $a_1<a_2<\dots <a_{n-1}$ with sum $\sum_{i=1}^{n-1}a_i = 0$, and consider $n-1$ functions $f_1,f_2,...f_{n-1}:\mathbb{R}\rightarrow \mathbb{R}$, each of them $n-2$ times differentiable over $\mathbb{R}$. Prove that there exists $a\in (a_1,a_{n-1})$ and $\theta, \theta_1,...,\theta_{n-1}\in \mathbb{R}$, not all zero, such that $$\sum_{k=1}^{n-1} \theta_k a_k=\theta a$$ and, at the same time, $$\sum_{k=1}^{n-1}\theta_kf_i(a_k)=\theta f_i^{(n-2)}(a)$$ for all $i\in\{1,2...,n-1\} $. \\ \\ [i](Sergiu Novac)[/i]

Determine twice differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which verify relation \[ \left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}. \]

Let $f:[0, \infty) \to (0, \infty)$ be an increasing function and $g:[0, \infty) \to \mathbb{R}$ be a two times differentiable function such that $g''$ is continuous and $g''(x)+f(x)g(x) = 0, \: \forall x \geq 0.$ $\textbf{a)}$ Provide an example of such functions, with $g \neq 0.$ $\textbf{b)}$ Prove that $g$ is bounded.