2015-12-1

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Problema:} Fie $f:[-1,1]\to \mathbb {R}$ o funcție crescătoare, derivabilă pe $[-1,1]$ cu $f'(0)\neq 0$. Să se arate ca exista cel puțin un punct $c\in (-1,1),c\neq 0$, cu proprietatea că

$$2cf(c)+\int _{0}^{c}f(x)\,dx\geq 0$$

.

$Solutie:\ (Robert\ Rogozsan)$ Daca $\ f(0)\geq 0$, cum $f$ e crescătoare, vom avea ca $f(t)\geq 0,\forall t\geq 0$, deci $2tf(t)+\int _{0}^{t}f(x)\,dx\geq 0,\forall t\geq 0$. Atunci luam $c\in (0,1)$ arbitrar si concluzia este verificata. Analog pentru $\ f(0)\leq 0$ (luam $c$ din $(-1,0)$).

$Observatie:$ In functie de cum e $f(0)$ fata de Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} , concluzia se verifica pentru Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle orice \ c \in (0,1)} (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-1,0)} ). Nu avem nevoie de faptul ca Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} e derivabila, nici de Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f'(0) \neq 0} .