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 sobre a função
. Precisamos encontrar vários aspectos dessa função, como pontos críticos, intervalos de crescimento e decrescimento, máximos e mínimos, concavidade, pontos de inflexão, assíntotas e, por fim, desenhar o gráfico. Para isso, vamos usar cálculos de derivadas, que nos ajudam a entender o comportamento da função.
Passo 2
Primeiro, vamos encontrar os pontos críticos da função. Pontos críticos são onde a derivada da função é zero ou não existe. Para isso, precisamos calcular a primeira derivada
.
A derivada de uma função do tipo
é dada pela regra do quociente:
Onde
e
. Vamos calcular
e
:
Substituindo na fórmula da derivada:
Agora, igualamos
a zero para encontrar os pontos críticos:
Resolvendo:
Portanto, o único ponto crítico real é
.
Passo 3
Agora, vamos descobrir os intervalos de crescimento e decrescimento da função. Isso é feito analisando o sinal da primeira derivada
.
Sabemos que:
O denominador
é sempre positivo, então o sinal de
depende apenas do numerador
.
Vamos analisar:
Como
e
para todo
, o numerador é sempre positivo. Assim,
para todo
, exceto em
onde é zero.
Portanto:
- A função é crescente em todo o seu domínio.
Passo 4
Vamos identificar agora os pontos de máximo e mínimo. Para isso, usamos a primeira derivada e os pontos críticos encontrados.
Já vimos que a única crítica é
. Vamos verificar o comportamento da função nesse ponto.
Como a derivada é positiva em todos os lados de
e igual a zero somente em
, isso indica que a função não tem máximo nem mínimo local nesse ponto. Na verdade,
é um ponto de inflexão, mas confirmaremos isso no próximo passo.
Passo 5
Agora, vamos determinar os intervalos de concavidade e os possíveis pontos de inflexão. Para isso, precisamos da segunda derivada
.
Primeiro, vamos calcular a segunda derivada a partir da primeira:
Vamos derivar isso usando a regra do quociente novamente:
Isso pode ser simplificado, mas para análise de sinal, vamos focar no numerador:
Após simplificar, encontramos:
Então:
Simplificando:
Agora, vamos analisar o sinal da segunda derivada:
Os pontos críticos para a concavidade são onde
ou não existe:
Vamos criar uma tabela de sinais para
:
- Para
: , → → Concavidade para cima - Para
: , → → Concavidade para baixo - Para
: , → → Concavidade para cima - Para
: , → → Concavidade para baixo
Passo 6
Com a análise da segunda derivada, podemos identificar os pontos de inflexão, que são os pontos onde a concavidade da função muda. Observamos que isso acontece em
,
e
.
Vamos verificar o comportamento nesses pontos:
- Em
: muda de concavidade para cima para para baixo. - Em
: muda de concavidade para baixo para para cima. - Em
: muda de concavidade para cima para para baixo.
Portanto, os pontos de inflexão são
,
e
.
Passo 7
Vamos agora identificar possíveis assíntotas da função
.
-
Assíntota Vertical: Ocorre onde o denominador é zero, a menos que o numerador também seja zero. Aqui,
não tem solução real, então não há assíntotas verticais. -
Assíntota Horizontal: Analisamos o limite quando
tende ao infinito:
Como tende ao infinito, não há assíntota horizontal.
- Assíntota Oblíqua: Para funções onde o grau do numerador é exatamente um a mais que o denominador, há uma assíntota oblíqua. Aqui, grau do numerador é 3 e do denominador é 2, então existe uma assíntota oblíqua.
Fazendo a divisão polinomial:
Portanto, a assíntota oblíqua é
.
Resposta
A função
apresenta:
a) Ponto crítico em
.
b) Intervalo de crescimento: toda a reta real
.
c) Não possui pontos de máximo ou mínimo locais.
d) Concavidade:
- Para
: para cima. - Para
: para baixo. - Para
: para cima. - Para
: para baixo.
e) Pontos de inflexão em
,
e
.
f) Assíntota oblíqua:
.
g) Gráfico: O gráfico da função é uma curva contínua sem assíntotas verticais ou horizontais, aproximando-se da linha
para valores grandes de
, com mudanças na concavidade nos pontos de inflexão identificados.
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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</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 ">
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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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</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=“15.79ex” height=“2.473ex” role=“img” focusable=“false” viewBox=“0 -843 6979 1093”><title>h(x) = 2 * raiz(x) - 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=“mn” transform=“translate(3259.6,0)”><path data-c=“32” d=“M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 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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=“15.79ex” height=“2.473ex” role=“img” focusable=“false” viewBox=“0 -843 6979 1093”><title>h(x) = 2 * raiz(x) - 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 ">
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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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