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Considere a função \(f(x) = \frac {2x^2}{x^2-1}\).

O domínio de \(f\) (da esquerda para a direita) é \(\left (\class{xm-answer}{\cssId{xm-answer-1}{\phantom{\rule{7em}{1.5em}}}},\class{xm-answer}{\cssId{xm-answer-2}{\phantom{\rule{7em}{1.5em}}}}\right )\cup \left (\class{xm-answer}{\cssId{xm-answer-3}{\phantom{\rule{7em}{1.5em}}}},\class{xm-answer}{\cssId{xm-answer-4}{\phantom{\rule{7em}{1.5em}}}}\right )\cup \left (\class{xm-answer}{\cssId{xm-answer-5}{\phantom{\rule{7em}{1.5em}}}},\class{xm-answer}{\cssId{xm-answer-6}{\phantom{\rule{7em}{1.5em}}}}\right )\).

\(f\) tem uma assíntota horizontal em \(y=\class{xm-answer}{\cssId{xm-answer-7}{\phantom{\rule{7em}{1.5em}}}}\).

\(f\) tem assíntotas verticais (da esquerda para a direita) em \(x=\class{xm-answer}{\cssId{xm-answer-8}{\phantom{\rule{7em}{1.5em}}}}\) e \(x=\class{xm-answer}{\cssId{xm-answer-9}{\phantom{\rule{7em}{1.5em}}}}\).

A derivada de \(f\) é dada por \[ f'(x) =\frac { \class{xm-answer}{\cssId{xm-answer-10}{\phantom{\rule{7em}{1.5em}}}}}{(x^2-1)^2}. \]

A segunda derivada de \(f\) é dada por \[ f''(x) =\frac { \class{xm-answer}{\cssId{xm-answer-11}{\phantom{\rule{7em}{1.5em}}}}}{(x^2-1)^3}. \]

\(f\) é crescente e tem concavidade para cima em \(\left (\class{xm-answer}{\cssId{xm-answer-12}{\phantom{\rule{7em}{1.5em}}}},\class{xm-answer}{\cssId{xm-answer-13}{\phantom{\rule{7em}{1.5em}}}}\right )\).

\(f\) é crescente e tem concavidade para baixo em \(\left (\class{xm-answer}{\cssId{xm-answer-14}{\phantom{\rule{7em}{1.5em}}}},\class{xm-answer}{\cssId{xm-answer-15}{\phantom{\rule{7em}{1.5em}}}}\right )\).

\(f\) é decrescente e tem concavidade para cima em \(\left (\class{xm-answer}{\cssId{xm-answer-16}{\phantom{\rule{7em}{1.5em}}}},\class{xm-answer}{\cssId{xm-answer-17}{\phantom{\rule{7em}{1.5em}}}}\right )\).

\(f\) é decrescente e tem concavidade para baixo em \(\left (\class{xm-answer}{\cssId{xm-answer-18}{\phantom{\rule{7em}{1.5em}}}},\class{xm-answer}{\cssId{xm-answer-19}{\phantom{\rule{7em}{1.5em}}}}\right )\).

\(f\) tem um máximo local em \(x=\class{xm-answer}{\cssId{xm-answer-20}{\phantom{\rule{7em}{1.5em}}}}\).

\(f\) tem \(\class{xm-answer}{\cssId{xm-answer-21}{\phantom{\rule{7em}{1.5em}}}}\) mínimos locais.

\(f\) temnão tem pontos de inflexão em \(x=-1\) e \(x=1\).

Considere os gráficos abaixo.

[Picture][Picture]

[Picture][Picture]

Escolha o gráfico que representa corretamente \(f(x)\).

\(A\) \(B\) \(C\) \(D\)