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Les vendes anuals S d'un nou producte s'expressen segons aquesta fórmula: S equals fraction numerator 5000 space t squared over denominator 8 space plus space t squared end fraction space space space space space space space o n space space space space space 0 less or equal than t less or equal than 3 space space space space space space space space i space space space space t equals e x p r e s s a t space e n space a n y s

a) Completa la taula següent



   x      0.5     
     1  
    1.5  
     2  
     2.5        3
  S



b)  En quin moment concret les vendes van ser les màximes?


Resposta

S left parenthesis 0 comma 5 right parenthesis equals fraction numerator 5000 space times left parenthesis 0 comma 5 right parenthesis squared over denominator 8 space plus space left parenthesis 0 comma 5 right parenthesis squared end fraction space space space space equals space 151 space a r t i c l e s space v e n u t s S left parenthesis 1 right parenthesis equals fraction numerator 5000 space times left parenthesis 1 right parenthesis squared over denominator 8 space plus space left parenthesis 1 right parenthesis squared end fraction space space space space equals space 555 space a r t i c l e s space v e n u t s S left parenthesis 1 comma 5 right parenthesis equals fraction numerator 5000 space times left parenthesis 1 comma 5 right parenthesis squared over denominator 8 space plus space left parenthesis 1 comma 5 right parenthesis squared end fraction space space space space equals space 1097 space a r t i c l e s space v e n u t s S left parenthesis 2 right parenthesis equals fraction numerator 5000 space times left parenthesis 2 right parenthesis squared over denominator 8 space plus space left parenthesis 2 right parenthesis squared end fraction space space space space equals space 1666 space a r t i c l e s space v e n u t s S left parenthesis 2 comma 5 right parenthesis equals fraction numerator 5000 space times left parenthesis 2 comma 5 right parenthesis squared over denominator 8 space plus space left parenthesis 2 comma 5 right parenthesis squared end fraction space space space space equals space 2192 space a r t i c l e s space v e n u t s S left parenthesis 3 right parenthesis equals fraction numerator 5000 space times left parenthesis 3 right parenthesis squared over denominator 8 space plus space left parenthesis 3 right parenthesis squared end fraction space space space space equals space 2647 space a r t i c l e s space v e n u t s


Ja es veu que a mida que va passant el temps, va creixent el nombre d'articles venuts. En l'interval [0,3] sembla que el màxim s'obtindrà en x=3

Estudiem la derivada per saber si hi ha un extrem en l'interval [0,3]. Si no hi ha cap extrem, vol dir que el mínim i màxim de la funció serà o en x=0 o en x=3

S apostrophe left parenthesis x right parenthesis space equals space fraction numerator open parentheses 10000 space t space close parentheses times space left parenthesis 8 space plus space t squared right parenthesis minus open parentheses 5000 space t squared close parentheses times open parentheses 2 t close parentheses over denominator space left parenthesis 8 space plus space t squared right parenthesis squared end fraction equals fraction numerator 80000 space t plus 10000 space t cubed minus 10000 space t cubed over denominator space left parenthesis 8 space plus space t squared right parenthesis squared end fraction S apostrophe left parenthesis x right parenthesis space equals fraction numerator 80000 space t over denominator space left parenthesis 8 space plus space t squared right parenthesis squared end fraction S apostrophe left parenthesis x right parenthesis equals 0 fraction numerator 80000 space t over denominator space left parenthesis 8 space plus space t squared right parenthesis squared end fraction equals 0 rightwards double arrow t equals 0


Veiem que si hi ha un extrem (màxim o mínim) relatiu ha d'estar en x=0

S(0) = 0 articles. Correspon al mínim

El màxim per tant està  en x=3.


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