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Justifiqueu que la funció polinòmica f(x) = x3 + x + 1 té un zero comprés entre −1 y 0.

O sigui que aquesta equació x3 + x + 1=0 té una solució, el valor de la qual està entre -1 i 0

Aproximeu aquest valor a 1 decimal.


Resposta:


Buscar el zero d'una funció f(x) és equivalent a resoldre l'equació f(x)=0 i trobar la solució.

A vegades trobar una solució d'una equació és difícil, per això en ocasions es pot usar el T. de Bolzano, i  tractar de "cercar-la"

De l'enunciat es pot pensar que la solució està en  [-1,0], i per tant serà aproximadament : x= -1 o bé x≈-0,9  o bé  x≈-0,8   o bé    x≈-0,7 ....   o bé   x≈-0,1 o bé x=0

Apliquem el T. de Bolzano en l'interval [-1,0]

begin mathsize 12px style space space space open table row cell table row cell table row cell table row cell table row cell f left parenthesis x right parenthesis space é s space u n a space f u n c i ó space p o l i n ò m i c a space i space p e r space t a n t space é s space c o n t i n u a space e n space e l space i n t e r v a l space left square bracket negative 1 comma space 0 right square bracket end cell row cell f left parenthesis negative 1 right parenthesis space equals space left parenthesis negative 1 right parenthesis cubed space plus space left parenthesis negative 1 right parenthesis space plus space 1 space equals space minus 1 space less than space 0 end cell row cell f left parenthesis 0 right parenthesis space equals space 0 space plus space 0 space plus space 1 greater than 0 end cell end table end cell end table end cell end table end cell end table end cell end table close curly brackets rightwards double arrow P e l space T. B o l z a n o space e x i s t e i x space u n space v a l o r space " c " space q u e space space c space element of space left parenthesis negative 1 comma space 0 right parenthesis space t a l space q u e colon space f left parenthesis c right parenthesis space equals space 0 end style

Per tant segur que hi ha (al menys) un valor que és solució de l'equació i que aquest valor és x≈-0,9  o bé  x≈-0,8   o bé    x≈-0,7 ....   o bé   x≈-0,1


Per afinar més i saber el primer decimal de la solució hem de fer el punt mig de l'interval  [−1, -0.5]

begin mathsize 12px style space space space open table row cell table row cell table row cell table row cell table row cell f left parenthesis x right parenthesis space é s space u n a space f u n c i ó space p o l i n ò m i c a space i space p e r space t a n t space é s space c o n t i n u a space e n space e l space i n t e r v a l space left square bracket negative 1 comma space minus 0.5 right square bracket end cell row cell f left parenthesis negative 1 right parenthesis space equals space left parenthesis negative 1 right parenthesis cubed space plus space left parenthesis negative 1 right parenthesis space plus space 1 space equals space minus 1 space less than space 0 end cell row cell f left parenthesis negative 0.5 right parenthesis space equals space left parenthesis negative 0.5 right parenthesis cubed space plus space left parenthesis negative 0.5 right parenthesis space plus space 1 equals 0.375 greater than 0 end cell end table end cell end table end cell end table end cell end table end cell end table close curly brackets rightwards double arrow P e l space T. B o l z a n o space e x i s t e i x space u n space v a l o r space " c " space q u e space space c space element of space left parenthesis negative 1 comma negative 0.5 right parenthesis space t a l space q u e colon space f left parenthesis c right parenthesis space equals space 0 end style



 Per tant x≈-0,9  o bé  x≈-0,8   o bé    x≈-0,7 ....   o bé   x≈-0,5


Seguim, tornem a dividir l'interval en dues parts i provarem en l'interval [−1, -0.8]. I si no funciona buscarem en l'interval [−0,8, -0.5]

begin mathsize 12px style space space space open table row cell table row cell table row cell table row cell table row cell f left parenthesis x right parenthesis space é s space u n a space f u n c i ó space p o l i n ò m i c a space i space p e r space t a n t space é s space c o n t i n u a space e n space e l space i n t e r v a l space left square bracket negative 1 comma space minus 0.8 right square bracket end cell row cell f left parenthesis negative 1 right parenthesis space equals space left parenthesis negative 1 right parenthesis cubed space plus space left parenthesis negative 1 right parenthesis space plus space 1 space equals space minus 1 space less than space 0 end cell row cell f left parenthesis negative 0.8 right parenthesis space equals space left parenthesis negative 0.8 right parenthesis cubed space plus space left parenthesis negative 0.8 right parenthesis space plus space 1 equals negative 0.312 less than 0 end cell end table end cell end table end cell end table end cell end table end cell end table close curly brackets rightwards double arrow N o space p o d e m space a p l i c a r space e l space T. space d e space B o l z a n o end style



Provem en [-0,8,-0.5]
begin mathsize 12px style space space space open table row cell table row cell table row cell table row cell table row cell f left parenthesis x right parenthesis space é s space u n a space f u n c i ó space p o l i n ò m i c a space i space p e r space t a n t space é s space c o n t i n u a space e n space e l space i n t e r v a l space left square bracket negative 0.8 comma space minus 0.5 right square bracket end cell row cell f left parenthesis negative 0.8 right parenthesis space equals space left parenthesis negative 0.8 right parenthesis cubed space plus space left parenthesis negative 0.8 right parenthesis space plus space 1 equals negative 0.312 less than 0 end cell row cell f left parenthesis negative 0.5 right parenthesis space equals space left parenthesis negative 0.5 right parenthesis cubed space plus space left parenthesis negative 0.5 right parenthesis space plus space 1 equals 0.375 greater than 0 end cell end table end cell end table end cell end table end cell end table end cell end table close curly brackets rightwards double arrow P e l space T. B o l z a n o space e x i s t e i x space u n space v a l o r space " c " space q u e space space c space element of space left parenthesis negative 0.8 comma negative 0.5 right parenthesis space t a l space q u e colon space f left parenthesis c right parenthesis space equals space o end style
Per tant x≈-0,7  o bé  x≈-0,6   o bé   x≈-0,5


Provem en [-0,8,-0.6]
begin mathsize 12px style space space space open table row cell table row cell table row cell table row cell table row cell f left parenthesis x right parenthesis space é s space u n a space f u n c i ó space p o l i n ò m i c a space i space p e r space t a n t space é s space c o n t i n u a space e n space e l space i n t e r v a l space left square bracket negative 0.8 comma space minus 0.6 right square bracket end cell row cell f left parenthesis negative 0.8 right parenthesis space equals space left parenthesis negative 0.8 right parenthesis cubed space plus space left parenthesis negative 0.8 right parenthesis space plus space 1 equals negative 0.312 less than 0 end cell row cell f left parenthesis negative 0.6 right parenthesis space equals space left parenthesis negative 0.6 right parenthesis cubed space plus space left parenthesis negative 0.6 right parenthesis space plus space 1 equals 0.184 greater than 0 end cell end table end cell end table end cell end table end cell end table end cell end table close curly brackets rightwards double arrow P e l space T. B o l z a n o space e x i s t e i x space u n space v a l o r space " c " space q u e space space c space element of space left parenthesis negative 0.8 comma negative 0.6 right parenthesis space t a l space q u e colon space f left parenthesis c right parenthesis space equals space o end style
Per tant x≈-0,7  o bé  x≈-0,6

Encara hauríem de provar en un interval [-0,8,-0.7]

begin mathsize 12px style space space space open table row cell table row cell table row cell table row cell table row cell f left parenthesis x right parenthesis space é s space u n a space f u n c i ó space p o l i n ò m i c a space i space p e r space t a n t space é s space c o n t i n u a space e n space e l space i n t e r v a l space left square bracket negative 0.8 comma space minus 0.7 right square bracket end cell row cell f left parenthesis negative 0.8 right parenthesis space equals space left parenthesis negative 0.8 right parenthesis cubed space plus space left parenthesis negative 0.8 right parenthesis space plus space 1 equals negative 0.312 less than 0 end cell row cell f left parenthesis negative 0.7 right parenthesis space equals space left parenthesis negative 0.7 right parenthesis cubed space plus space left parenthesis negative 0.7 right parenthesis space plus space 1 equals negative 0.043 less than 0 end cell end table end cell end table end cell end table end cell end table end cell end table close curly brackets rightwards double arrow E l space T. B o l z a n o space n o space f u n c i o n a end style

Hauríem de provar en un interval [-0,7,-0.6]
begin mathsize 12px style space space space open table row cell table row cell table row cell table row cell table row cell f left parenthesis x right parenthesis space é s space u n a space f u n c i ó space p o l i n ò m i c a space i space p e r space t a n t space é s space c o n t i n u a space e n space e l space i n t e r v a l space left square bracket negative 0.7 comma space minus 0.6 right square bracket end cell row cell f left parenthesis negative 0.7 right parenthesis space equals space left parenthesis negative 0.7 right parenthesis cubed space plus space left parenthesis negative 0.7 right parenthesis space plus space 1 equals negative 0.043 less than 0 end cell row cell f left parenthesis negative 0.6 right parenthesis space equals space left parenthesis negative 0.6 right parenthesis cubed space plus space left parenthesis negative 0.6 right parenthesis space plus space 1 equals 0.184 greater than 0 end cell end table end cell end table end cell end table end cell end table end cell end table close curly brackets rightwards double arrow P e l space T. B o l z a n o space e x i s t e i x space u n space v a l o r space " c " space q u e space space c space element of space left parenthesis negative 0.7 comma negative 0.6 right parenthesis space t a l space q u e colon space f left parenthesis c right parenthesis space equals space o end style

Per tant   x≈-0,6

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