Responda os seguintes itens das funções dadas abaixo:
a) Encontre o(s) ponto(s) crítico(s), se existirem,
b) Intervalos de Crescimento e Decrescimento, se existirem,
c) Ponto(s) máximo(s) e/ou mínimo(s), se existirem,
d) Intervalos de Concavidade para Cima e para Baixo, se existirem,
e) Ponto(s) inflexão, se existirem,
f) Assíntota(s), se existirem,
g) Construa o gráfico.
a) Encontre o(s) ponto(s) crítico(s), se existirem,
b) Intervalos de Crescimento e Decrescimento, se existirem,
c) Ponto(s) máximo(s) e/ou mínimo(s), se existirem,
d) Intervalos de Concavidade para Cima e para Baixo, se existirem,
e) Ponto(s) inflexão, se existirem,
f) Assíntota(s), se existirem,
g) Construa o gráfico.
Passo 1
Beleza, vamos resolver o exercício que pede para analisar a função
. Vamos usar conceitos de cálculo, como derivadas, para encontrar pontos críticos, intervalos de crescimento e decrescimento, máximos e mínimos, concavidade, pontos de inflexão, assíntotas e, por fim, construir o gráfico da função. Vamos precisar calcular primeiras e segundas derivadas e entender o comportamento da função com base nelas.
Passo 2
a) Encontrar o(s) ponto(s) crítico(s)
Pontos críticos são onde a primeira derivada da função é zero ou não existe. Vamos calcular a primeira derivada de
.
Primeira derivada:
Agora, igualamos
a zero:
Isso ocorre quando:
Então, os pontos críticos são
,
e
.
Passo 3
b) Intervalos de Crescimento e Decrescimento
Vamos analisar o sinal da primeira derivada
nos intervalos definidos pelos pontos críticos
,
e
.
-
Intervalo
:
Escolha: Crescente. -
Intervalo
:
Escolha: Crescente. -
Intervalo
:
Escolha: Decrescente. -
Intervalo
:
Escolha: Decrescente.
Passo 4
c) Ponto(s) máximo(s) e/ou mínimo(s)
Vamos usar a análise dos intervalos de crescimento e decrescimento:
-
Em
:
A derivada passa de positiva para positiva, então não há máximo ou mínimo. -
Em
:
A derivada passa de positiva para negativa, indicando um máximo em. -
Em
:
A derivada passa de negativa para negativa, então não há máximo ou mínimo.
Passo 5
d) Intervalos de Concavidade para Cima e para Baixo
Para determinar a concavidade, precisamos da segunda derivada
.
Calculando a segunda derivada:
Agora, analisamos o sinal de
:
-
Quando
(ou seja, ):
O termoé positivo. Então, a concavidade depende de: - Para
: → Concavidade para baixo. - Para
: → Concavidade para cima.
- Para
-
Quando
(ou seja, ):
O termoé negativo. Epara , então:
Passo 6
e) Ponto(s) inflexão
Pontos de inflexão ocorrem onde a concavidade muda, ou seja, onde a segunda derivada muda de sinal.
Analisando
:
- Mudança de concavidade em
e .
Verificando:
- Em
: Mudança de concavidade de para baixo para para cima. - Em
: A função pode ter pontos de inflexão, mas como é definida para todos os valores reais, esses pontos também são pontos de inflexão.
Então, os pontos de inflexão são
e
.
Passo 7
f) Assíntota(s)
Assíntotas são linhas que a função se aproxima mas nunca toca ou cruza. Vamos verificar as possíveis assíntotas da função
.
Como a função é um polinômio (pois
expande para um polinômio), não possui assíntotas verticais ou horizontais.
Passo 8
g) Construir o gráfico
Com todas as informações obtidas:
- Pontos críticos em
, , . Máximo em . - Crescimento em
e . - Decrescimento em
e . - Concavidade para baixo em
e , e para cima em . - Pontos de inflexão em
e . - Sem assíntotas.
Com esses dados, podemos esboçar o gráfico, identificando os comportamentos nos intervalos e pontos críticos.
Resposta
A função
apresenta pontos críticos em
,
e
, com um máximo local em
. Cresce nos intervalos
e
, e decresce em
e
. Possui concavidade para baixo em
e
, e para cima em
. Os pontos de inflexão estão em
e
. Não há assíntotas. O gráfico reflete esses comportamentos, com a forma simétrica característica de uma função par elevada a uma potência ímpar.
Exercícios relacionados
<html><head></head><body><div>Responda os seguintes itens das funções dadas abaixo:<br>
a) Encontre o(s) ponto(s) crítico(s), se existirem,<br>
b) Intervalos de Crescimento e Decrescimento, se existirem,<br>
c) Ponto(s) máximo(s) e/ou mínimo(s), se existirem,<br>
d) Intervalos de Concavidade para Cima e para Baixo, se existirem,<br>
e) Ponto(s) inflexão, se existirem,<br>
f) Assíntota(s), se existirem,<br>
g) Construa o gráfico.<br>
<span class=“math-inline “>
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</body></html>
a) Encontre o(s) ponto(s) crítico(s), se existirem,<br>
b) Intervalos de Crescimento e Decrescimento, se existirem,<br>
c) Ponto(s) máximo(s) e/ou mínimo(s), se existirem,<br>
d) Intervalos de Concavidade para Cima e para Baixo, se existirem,<br>
e) Ponto(s) inflexão, se existirem,<br>
f) Assíntota(s), se existirem,<br>
g) Construa o gráfico.<br>
<span class=“math-inline “>
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<html><head></head><body><div>Responda os seguintes itens das funções dadas abaixo:<br>
a) Encontre o(s) ponto(s) crítico(s), se existirem,<br>
b) Intervalos de Crescimento e Decrescimento, se existirem,<br>
c) Ponto(s) máximo(s) e/ou mínimo(s), se existirem,<br>
d) Intervalos de Concavidade para Cima e para Baixo, se existirem,<br>
e) Ponto(s) inflexão, se existirem,<br>
f) Assíntota(s), se existirem,<br>
g) Construa o gráfico.<br>
<span class="math-inline ">
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</body></html>
a) Encontre o(s) ponto(s) crítico(s), se existirem,<br>
b) Intervalos de Crescimento e Decrescimento, se existirem,<br>
c) Ponto(s) máximo(s) e/ou mínimo(s), se existirem,<br>
d) Intervalos de Concavidade para Cima e para Baixo, se existirem,<br>
e) Ponto(s) inflexão, se existirem,<br>
f) Assíntota(s), se existirem,<br>
g) Construa o gráfico.<br>
<span class="math-inline ">
<mjx-container class=“MathJax” jax=“SVG”><svg style=“vertical-align: -0.566ex;” xmlns=“http://www.w3.org/2000/svg” width=“15.531ex” height=“2.452ex” role=“img” focusable=“false” viewBox=“0 -833.9 6864.5 1083.9”><title>h(x) = e^(2x) - e^x</title><g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“scale(1,-1)”><g data-mml-node=“math”><g data-mml-node=“mi”><path data-c=“210E” d=“M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z”></path></g><g data-mml-node=“mo” transform=“translate(576,0)”><path 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</body></html>
<html><head></head><body><div>Responda os seguintes itens das funções dadas abaixo:<br>
a) Encontre o(s) ponto(s) crítico(s), se existirem,<br>
b) Intervalos de Crescimento e Decrescimento, se existirem,<br>
c) Ponto(s) máximo(s) e/ou mínimo(s), se existirem,<br>
d) Intervalos de Concavidade para Cima e para Baixo, se existirem,<br>
e) Ponto(s) inflexão, se existirem,<br>
f) Assíntota(s), se existirem,<br>
g) Construa o gráfico.<br>
<span class="math-inline ">
<mjx-container class=“MathJax” jax=“SVG”><svg style=“vertical-align: -0.566ex;” xmlns=“http://www.w3.org/2000/svg” width=“17.55ex” height=“2.473ex” role=“img” focusable=“false” viewBox=“0 -843 7757 1093”><title>h(x) = (x - 3) * raiz(x)</title><g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“scale(1,-1)”><g data-mml-node=“math”><g data-mml-node=“mi”><path data-c=“210E” d=“M137 683Q138 683 209 688T282 694Q294 694 294 685Q294 674 258 534Q220 386 220 383Q220 381 227 388Q288 442 357 442Q411 442 444 415T478 336Q478 285 440 178T402 50Q403 36 407 31T422 26Q450 26 474 56T513 138Q516 149 519 151T535 153Q555 153 555 145Q555 144 551 130Q535 71 500 33Q466 -10 419 -10H414Q367 -10 346 17T325 74Q325 90 361 192T398 345Q398 404 354 404H349Q266 404 205 306L198 293L164 158Q132 28 127 16Q114 -11 83 -11Q69 -11 59 -2T48 16Q48 30 121 320L195 616Q195 629 188 632T149 637H128Q122 643 122 645T124 664Q129 683 137 683Z”></path></g><g data-mml-node=“mo” transform=“translate(576,0)”><path 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289Z”></path></g><g data-mml-node=“mo” transform=“translate(1537,0)”><path data-c=“29” d=“M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z”></path></g><g data-mml-node=“mo” transform=“translate(2203.8,0)”><path data-c=“3D” d=“M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z”></path></g><g data-mml-node=“mo” transform=“translate(3259.6,0)”><path data-c=“28” d=“M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z”></path></g><g data-mml-node=“mi” transform=“translate(3648.6,0)”><path 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</body></html>
a) Encontre o(s) ponto(s) crítico(s), se existirem,<br>
b) Intervalos de Crescimento e Decrescimento, se existirem,<br>
c) Ponto(s) máximo(s) e/ou mínimo(s), se existirem,<br>
d) Intervalos de Concavidade para Cima e para Baixo, se existirem,<br>
e) Ponto(s) inflexão, se existirem,<br>
f) Assíntota(s), se existirem,<br>
g) Construa o gráfico.<br>
<span class="math-inline ">
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<html><head></head><body><div>Responda os seguintes itens das funções dadas abaixo:<br>
a) Encontre o(s) ponto(s) crítico(s), se existirem,<br>
b) Intervalos de Crescimento e Decrescimento, se existirem,<br>
c) Ponto(s) máximo(s) e/ou mínimo(s), se existirem,<br>
d) Intervalos de Concavidade para Cima e para Baixo, se existirem,<br>
e) Ponto(s) inflexão, se existirem,<br>
f) Assíntota(s), se existirem,<br>
g) Construa o gráfico.<br>
<span class="math-inline ">
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a) Encontre o(s) ponto(s) crítico(s), se existirem,<br>
b) Intervalos de Crescimento e Decrescimento, se existirem,<br>
c) Ponto(s) máximo(s) e/ou mínimo(s), se existirem,<br>
d) Intervalos de Concavidade para Cima e para Baixo, se existirem,<br>
e) Ponto(s) inflexão, se existirem,<br>
f) Assíntota(s), se existirem,<br>
g) Construa o gráfico.<br>
<span class="math-inline ">
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