Question

1) Indique se as funções de 2º grau a seguir tem ponto de máximo ou mínimo:
a) f(x) = x 2 - 7x + 1 b) f(x) = -3x 2 + 10x + 4

2) Determine o ponto de mínimo da função f(x) = x 2 - 5x + 6

3) Determine o ponto de máximo da função f(x) = -x 2 + 3x + 1

Answer 1

$\boxed{\begin{array}{l}\rm 1)\\\tt a)~\sf f(x)=x^2-7x+1\\\sf como~a=1>0\implies tem~ponto~m\acute inimo\\\tt b)~\sf f(x)=-3x^2+10x+4\\\sf a=-3<0\implies tem~ponto~m\acute aximo\end{array}}$

$\Large\boxed{\begin{array}{l}\rm 2)~\sf f(x)=x^2-5x+6\\\sf\Delta=b^2-4ac\\\sf\Delta=(-5)^2-4\cdot1\cdot6\\\sf\Delta=25-24\\\sf\Delta=1\\\sf x_V=-\dfrac{b}{2a}=-\dfrac{-5}{2\cdot1}=\dfrac{5}{2}\\\\\sf y_V=-\dfrac{\Delta}{4a}=-\dfrac{1}{4\cdot1}=-\dfrac{1}{4}\\\\\sf o~ponto~V(x_V,y_V)~\acute e~ponto~de~m\acute inimo\\\sf V\bigg(\dfrac{5}{2},-\dfrac{1}{4}\bigg)\end{array}}$

$\large\boxed{\begin{array}{l}\rm 3)\\\sf f(x)=-x^2+3x+1\\\sf\Delta=b^2-4ac\\\sf\Delta=3^2-4\cdot(-1)\cdot1\\\sf\Delta=9+4\\\sf\Delta=13\\\sf x_V=-\dfrac{b}{2a}=-\dfrac{3}{2\cdot(-1)}=\dfrac{3}{2}\\\\\sf y_V=-\dfrac{\Delta}{4a}=-\dfrac{13}{4\cdot(-1)}=\dfrac{13}{4}\\\\\sf O~ponto~V(x_V,y_V)~\acute e~o~ponto~de~m\acute aximo\\\sf V\bigg(\dfrac{3}{2},\dfrac{13}{4}\bigg)\end{array}}$