Question

As funções reais g(x)=x^2-b e f(x)=x-5/4 possuem um único ponto em comum. Qual o valor de b ?

Answer 1

$\mathrm{g(x)=x^2-b}\ \ \| \ \ \mathrm{f(x)=x-\cfrac{5}{4}}\\\\ \mathrm{Para\ g(x)=f(x),\ teremos:}\\\\ \mathrm{x^2-b=x-\cfrac{5}{4}\ \to\ x^2-x+\cfrac{5}{4}-b=0}\\\\ \mathrm{Para\ que\ g(x)\ e\ f(x)\ tenham\ um\ \'unico\ ponto}\\ \mathrm{em\ comum,\ \Delta\ deve\ ser\ igual\ a\ zero.\ Logo:} \\\\ \mathrm{\Delta=0\ \to\ (-1)^2-4.1.\bigg(\cfrac{5}{4}-b\bigg)=0}\\\\ \mathrm{1-5+4b=0\ \to\ -4+4b=0\ \to\ 4b=4}\\\\ \mathrm{b=\cfrac{4}{4}}\ \to\ \mathbf{b=1}$