28251

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 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} } 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 $f$ cu proprietățile din enunț.
b) Arătați că există $c\in [0,1]$ astfel încât $f(c)=c^{n^{3}-1}$.

Soluție. a) Funcția

$$f:[0,1]\longrightarrow \mathbb {R} ,\quad f(x)=\ln {\sqrt {1+{\frac {4x}{n^{3}}}}}$$

are toate proprietățile din enunț.

b) Deoarece $e^{t}\geq t+1$ pentru orice $t\in \mathbb {R}$, avem

$$1+{\frac {2}{n^{3}}}=\int _{0}^{1}e^{2f(x)}dx\geq \int _{0}^{1}\left(2f(x)+1\right)dx=2\int _{0}^{1}f(x)dx+1,$$

de unde rezultă că

$$\int _{0}^{1}f(x)dx\leq {\frac {1}{n^{3}}}.$$

Cum $\int _{0}^{1}x^{n^{3}-1}dx={\dfrac {1}{n^{3}}}$, deducem că $\int _{0}^{1}\left(f(x)-x^{n^{3}-1}\right)dx\leq 0$, deci există $a\in [0,1]$, 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 } . 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 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) \cdot g(a) \leq 0} .
Rezultă 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,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 } , ceea ce revine la faptul 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,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 f(c) = c^{n^{3}-1} } .