27036: Difference between revisions
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</math>, atunci <math>yf'(x)\geq 0, xf'(y)\geq0 | </math>, atunci <math>yf'(x)\geq 0, xf'(y)\geq0 | ||
</math> și ca mai sus avem <math>yf'(x)=xf'(y) | </math> și ca mai sus avem <math>yf'(x)=xf'(y) | ||
</math>. | </math>. | ||
În particular <math>f'(x)=-f'(-1)x | 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 | </math> și <math>d \in \mathbb{R} | ||
</math> | </math>. Cum <math>0 = f(0) = d, | ||
</math> rezultă că <math display="block">f(x) = bx^2, \, x \in \left(-\infty, 0\right]. | </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 | ||
</math>, deci <math>4b^3 = \frac{9}{4}b | </math>, deci <math>4b^3 = \frac{9}{4}b | ||
</math> și cum <math>b> 0 | |||
</math>, rezultă <math>b = \frac{3}{4}. | |||
</math> | </math> | ||
Obținem <math display="block">f(x) = \frac{3}{4}x^2, x \in \mathbb{R} | |||
</math>funcție care verifică ipotezele din enunț. | |||
</math> | |||
Revision as of 13:18, 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)
c) , oricare ar fi .
Soluție:
Cum , rezultă că este strict crescătoare, deci injectivă pe .
Deoarece expresia 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 xyf'(x)f'(y)-f(x)f(y) } este simetrică în variabilele 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 } 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 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 d \in \mathbb{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 \mathbb{R} } funcție care verifică ipotezele din enunț.