28251: Difference between revisions

Revision as of 18:50, 7 January 2024

28251 (Gheorghe Boroica)

Fie 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 (n \geq 2)} un număr natural și $f:[0,1]\longrightarrow \mathbb {R}$ o funcție continuă astfel încât 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) \geq 0} și 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 \int_{0}^{1} e^{2f(x)} dx = 1+\frac{2}{n^3}} .
a) Dați un exemplu de o funcție 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} cu proprietățile din enunț.
b) Arătați că există 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 c \in [0,1] } astfel încât 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(c) = c^{n^{3}} - 1 } .

Soluție. a) Funcția 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: [0,1] \longrightarrow \mathbb{R}, \quad f(x) = \ln\sqrt{1 + \frac{4x}{n^3}}} are toate proprietățile din enunț.
b) Deoarece 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 e^t \geq t + 1} pentru orice 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 t \in \mathbb{R}} , avem

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 + \frac{2}{n^3} = \int_{0}^{1} e^{2f(x)} dx \geq \int_{0}^{1} (2f(x) + 1) dx = 2\int_{0}^{1} f(x)dx + 1} ,
de unde rezultă că 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 \int_{0}^{1} f(x)dx\leq \frac{1}{n^3}} . Cum 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 \int_{0}^{1} x^{n^3-1}dx} , deducem că 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 \int_{0}^{1} (f(x) - x^{n^3-1}_dx \leq 0 } , deci există 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 a \in [0,1] } , astfel încât 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(a) - a^{n^{3}}-1 \leq 0 } .
Functia 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 g : [0,1] \longrightarrow \mathbb{R}, g(x) = f(x) - x^{n^{3}}} este continuă și 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 g(0) * g(a) \leq 0} .
Rezultă că exsită 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 c \in [0,a] \subseteq [0,1]} astfel încât 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 g(c) = 0 } .

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-'''28251  (Trif Flaviu) '''
+'''28251  (Gheorghe Boroica) '''
 <br />
 <br />
-''Fie'' <math>(n \geq 2)</math> ''un număr natural și'' <math> f:  [0,1] \longrightarrow  \mathbb{R} </math> ''o funcție continuă astfel încât'' <math>f(0) \geq 0</math> si <math>\int_{0}^{1} e^2f(x) dx = 1+\frac{2}{n^3}</math>.
+''Fie'' <math>(n \geq 2)</math> ''un număr natural și'' <math> f:  [0,1] \longrightarrow  \mathbb{R} </math> ''o funcție continuă astfel încât'' <math>f(0) \geq 0</math> și <math>\int_{0}^{1} e^{2f(x)} dx = 1+\frac{2}{n^3}</math>.
 <br />
-a) ''Dați un exemplu de o funcție f cu proprietățile din enunț''.
+a) ''Dați un exemplu de o funcție <math>f</math> cu proprietățile din enunț''.
 <br />
 b) ''Arătați că există'' <math> c \in [0,1] </math> astfel încât <math> f(c) = c^{n^{3}} - 1