27036: Difference between revisions
Pagină nouă: '''27036 (Radu Pop)''' ''Să se determine funcțiile derivabile <math>f </math> : ℝ -> ℝ cu proprietățile:'' ''a) <math>f' </math> este funcție strict crescătoare;'' ''b) <math>f'(0) = 0; </math>'' ''c) <math>f(yf'(x)) + f(x)f(y) = xy f'(x)f'(y) </math> , oricare ar fi x, y ∈ ℝ;'' '''Soluție:''' Cum <math>f' (x) > 0, x \in (0, \infty) </math>, rezultă că <math>f </math> este strict crescătoare, deci injectivă pe <math>[0, \infty) </math>. Deoarece e... |
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'''27036 (Radu Pop)''' | '''27036 (Radu Pop)''' | ||
''Să se determine funcțiile derivabile <math>f | ''Să se determine funcțiile derivabile <math>f : \mathbb{R} \to \mathbb{R} | ||
</math> | </math>'' ''cu proprietățile:'' | ||
''a) <math>f' | ''a) <math>f' | ||
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''c) <math>f(yf'(x)) + f(x)f(y) = xy f'(x)f'(y) | ''c) <math>f(yf'(x)) + f(x)f(y) = xy f'(x)f'(y) | ||
</math> , oricare ar fi x, y | </math> , oricare ar fi'' <math>x,y \in \mathbb{R} | ||
</math>. | |||
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</math>, rezultă că <math>f | </math>, rezultă că <math>f | ||
</math> este strict crescătoare, deci injectivă pe <math>[0, \infty) | </math> este strict crescătoare, deci injectivă pe <math>[0, \infty) | ||
</math>. Deoarece expresia <math>xyf'(x)f'(y)-f(x)f(y) | </math>. | ||
</math> simetrică | |||
</math>. Din injectivitatea | Deoarece expresia <math>xyf'(x)f'(y)-f(x)f(y) | ||
</math> este simetrică în variabilele <math>x | |||
</math> și <math>y | |||
</math>, din ipoteza c) rezultă că <math>f(yf'(x))=f(xf'(y)) | |||
</math>. Din injectivitatea funcției <math>f | |||
</math> obținem <math>yf'(x)=xf'(y) | </math> obținem <math>yf'(x)=xf'(y) | ||
</math>, pentru orice <math>x,y \in [0, \infty) | </math>, pentru orice <math>x,y \in [0, \infty) | ||
</math>. În particular, <math>f'(x) = f'(1)x | </math>. | ||
În particular, <math>f'(x) = f'(1)x | |||
</math> , deci <math>f(x) = ax^2 +c | </math> , deci <math>f(x) = ax^2 +c | ||
</math> , unde <math>a = | </math> , unde <math>a = \frac{f'(1)}{2} > 0 | ||
</math> și <math>c \in | </math> și <math>c \in | ||
</math> ℝ . Pentru <math>x, y \in [0, \infty) | </math> ℝ . Pentru <math>x, y \in [0, \infty) | ||
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</math> , deci <math>4a^3x^2y^2+a^2x^2y^2=4a^2x^2y^2 | </math> , deci <math>4a^3x^2y^2+a^2x^2y^2=4a^2x^2y^2 | ||
</math>. Rezultă <math>a = \frac{3}{4} | </math>. Rezultă <math>a = \frac{3}{4} | ||
</math> , deci <math>f(x) = \frac{3}{4}x^2, x \in [0, \infty) | </math> , deci <math display="block">f(x) = \frac{3}{4}x^2, x \in \left[0, \infty\right). | ||
</math> | </math>Dacă <math>x, y \in (-\infty, 0] | ||
</math>, atunci <math>yf'(x)\geq 0, xf'(y)\geq0 | </math>, atunci <math>yf'(x)\geq 0, xf'(y)\geq0 | ||
</math> și | </math> și ca mai sus avem <math>yf'(x)=xf'(y) | ||
</math>. În particular <math>f'(x)=-f'(-1)x | </math>. | ||
În particular <math>f'(x)=-f'(-1)x | |||
</math>, deci <math>f(x)=bx^2 + d, | </math>, deci <math>f(x)=bx^2 + d, | ||
</math> cu <math>b>0 | </math> cu <math>b>0 | ||
</math> și d <math>\in | </math> și d <math>\in | ||
</math> ℝ . Cum <math>0 = f(0) = d, | </math> ℝ . Cum <math>0 = f(0) = d, | ||
</math> rezultă că <math>f(x) = bx^2, x \in (-\infty, 0] | </math> rezultă că <math display="block">f(x) = bx^2, \, x \in \left(-\infty, 0\right]. | ||
</math> Pentru <math>x = -1, y = 1 | </math> Pentru <math>x = -1, y = 1 | ||
</math>, avem <math>f(-3b)+\frac{3}{4}b = 3b | </math>, avem <math>f(-3b)+\frac{3}{4}b = 3b | ||
Revision as of 13:14, 21 October 2023
27036 (Radu Pop)
Să se determine funcțiile derivabile 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 : \mathbb{R} \to \mathbb{R} } cu proprietățile:
a) 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' } este funcție strict crescătoare;
b) 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) = 0; }
c) 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(yf'(x)) + f(x)f(y) = xy f'(x)f'(y) } , oricare ar fi 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 x,y \in \mathbb{R} } .
Soluție:
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 f' (x) > 0, x \in (0, \infty) } , 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 f } este strict crescătoare, deci injectivă pe .
Deoarece expresia este simetrică în variabilele 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 y } , din ipoteza c) 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 f(yf'(x))=f(xf'(y)) } . Din injectivitatea funcției 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 } obținem 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 yf'(x)=xf'(y) } , 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 x,y \in [0, \infty) } .
În particular, 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'(x) = f'(1)x } , deci 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(x) = ax^2 +c } , unde 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 = \frac{f'(1)}{2} > 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 c \in } ℝ . 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 x, y \in [0, \infty) } 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 f(2axy)+a^2 x^2 y^2 = 4a^2 x^2y^2 } , deci 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 4a^3x^2y^2+a^2x^2y^2=4a^2x^2y^2 } . Rezultă 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 = \frac{3}{4} } , deci 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(x) = \frac{3}{4}x^2, x \in \left[0, \infty\right). } Dacă 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 x, y \in (-\infty, 0] } , atunci 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 yf'(x)\geq 0, xf'(y)\geq0 } și ca mai sus 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 yf'(x)=xf'(y) } .
În particular 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'(x)=-f'(-1)x } , deci 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(x)=bx^2 + d, } cu 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 b>0 } și d 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 \in } ℝ . 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 0 = f(0) = d, } 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 f(x) = bx^2, \, x \in \left(-\infty, 0\right]. } 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 x = -1, y = 1 } , 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 f(-3b)+\frac{3}{4}b = 3b } , deci 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 4b^3 = \frac{9}{4}b }
și 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 b> 0 } , rezultă 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 b = \frac{3}{4}. }
Obținem 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(x) = \frac{3}{4}x^2, x \in } ℝ , funcție care verifică ipotezele din enunț.