Question

considerem a funcao f(x)x^2+1.calcule os valores reais de x para que se tenha f(x+2)<f(2)

Answer 1

$\large\begin{array}{l} \textsf{Considerando a fun\c{c}\~ao}\\\\ \mathsf{f(x)=x^2+1}\\\\\\ \textsf{resolver a inequa\c{c}\~ao}\\\\ \mathsf{f(x+2)<f(2)}\\\\ \mathsf{(x+2)^2+\diagup\!\!\!\! 1<2^2+\diagup\!\!\!\! 1}\\\\ \mathsf{(x+2)^2<2^2}\\\\ \mathsf{(x+2)^2<4}\\\\ \mathsf{x^2+4x+\diagup\!\!\!\! 4<\diagup\!\!\!\! 4}\\\\ \mathsf{x^2+4x<0}\\\\ \mathsf{x\cdot (x+4)<0\qquad(i)} \end{array}$


$\large\begin{array}{l} \textsf{Agora, temos uma inequa\c{c}\~ao-produto. Vamos analisar os}\\\textsf{sinais de cada fator do lado esquerdo.}\\\\ \textsf{As ra\'izes do lado esquerdo s\~ao}\\\\ \mathsf{x_1=-4~~e~~x_2=0.}\\\\\\ \begin{array}{cc} \mathsf{x}~~&~~\underline{---}\underset{-4}{\bullet}\underline{----}\underset{0}{\bullet}\underline{+++}\\\\ \mathsf{x+4}~~&~~\underline{---}\underset{-4}{\bullet}\underline{++++}\underset{0}{\bullet}\underline{+++}\\\\ \mathsf{x\cdot (x+4)}~~&~~\underline{+++}\underset{-4}{\bullet}\underline{----}\underset{0}{\bullet}\underline{+++}\\\\ \end{array} \end{array}$


$\large\begin{array}{l} \textsf{Como queremos que o produto do lado esquerdo de (i)}\\\textsf{seja negativo, o intervalo de interesse \'e}\\\\ \mathsf{-4<x<0.}\\\\\\ \textsf{Conjunto solu\c{c}\~ao: }\mathsf{S=\{x\in \mathbb{R}:~-4<x<0\}}\\\\\\ \textsf{ou usando a nota\c{c}\~ao de intervalos,}\\\\ \mathsf{S=\left]-4,\,0\right[\,.} \end{array}$


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$\large\begin{array}{l} \textsf{D\'uvidas? Comente.}\\\\\\ \textsf{Bons estudos! :-)} \end{array}$


Tags: função quadrática segundo grau inequação produto composta