Question

Sabendo-se que o valor máximo da função f(x) = ax + bx + 4, é igual a 16, o qual ocorre em x = 4. O valor de b é igual a:

Answer 1

$\text{Dados da quest\~ao:}\\\\f(x)=ax^2+bx+4\qquad\qquad{f}(4)=16\qquad\qquad(x_v,y_v)=(4,\,16)\\\\b=\,?\\\\\\\text{Resolu\c{c}\~ao:}\\\\\begin{lcr}f(x)=ax^2+bx+4\\\\f(4)=16\Longrightarrow16=a(4)^2+b(4)+4\\\\16=8a+4b+4\\\\8a=16-4b-4\\\\a=\dfrac{12-4b}{8}\\\\a=\dfrac{3-b}{2}\qquad(\!1\!)\end{lcr}\\\\\\\\\\\begin{lcr}x_v=-\dfrac{b}{2a}\\\\4=-\dfrac{b}{2a}\\\\8a=-b\\\\a=-\dfrac{b}{8}\qquad(\!2\!)\end{lcr}\\\\\\\\\\\text{Igualando}\,\,(\!1\!)\,\,\text{e}\,\,(\!2\!)\text{:}\\\\\dfrac{3-b}{4}=-\dfrac{b}{8}\\\\\dfrac{8(3-b)}{4}=-b\\\\2(3-b)=-b\\\\6-2b=-b$

$6=2b-b\\\\\boxed{\boxed{6=b}}$