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
. A gente vai analisar diferentes características dessa função, como pontos críticos, intervalos de crescimento e decrescimento, máximos e mínimos, concavidade, pontos de inflexão, assíntotas e, por fim, construir o gráfico. Para isso, vamos usar conceitos de cálculo, especialmente derivadas.
Passo 2
a) Encontrar o(s) ponto(s) crítico(s), se existirem.
Primeiro, pontos críticos são onde a derivada da função é zero ou não existe. Vamos encontrar a primeira derivada de
.
Agora, igualamos a derivada a zero para achar os pontos críticos:
Podemos simplificar a equação dividindo todos os termos por -3:
Resolvendo a equação quadrática:
Portanto, temos dois pontos críticos:
Passo 3
b) Intervalos de Crescimento e Decrescimento, se existirem.
Para determinar onde a função está crescendo ou decrescendo, analisamos o sinal da primeira derivada
.
Os pontos críticos dividem a reta real em intervalos. Temos os pontos
e
.
Vamos testar um valor em cada intervalo:
-
Intervalo
: Escolha -
Intervalo
: Escolha -
Intervalo
: Escolha
Passo 4
c) Ponto(s) máximo(s) e/ou mínimo(s), se existirem.
Os pontos críticos podem ser máximos ou mínimos. Usamos o teste da primeira derivada para determinar isso:
-
No
:
A derivada muda de negativa para positiva quando passa por, indicando um mínimo local. -
No
:
A derivada muda de positiva para negativa quando passa por, indicando um máximo local.
Para encontrar as coordenadas, substituímos
na função original:
-
Mínimo em
: Então, o ponto mínimo é. -
Máximo em
: Então, o ponto máximo é.
Passo 5
d) Intervalos de Concavidade para Cima e para Baixo, se existirem.
Para analisar a concavidade, vamos calcular a segunda derivada da função:
Agora, determinamos onde
é positiva ou negativa:
Passo 6
e) Ponto(s) inflexão, se existirem.
Pontos de inflexão ocorrem onde a concavidade muda, ou seja, onde a segunda derivada é zero ou não existe.
Igualamos
a zero:
Agora, verificamos se há mudança de concavidade em
:
- Para
, (concavidade para cima) - Para
, (concavidade para baixo)
Como há mudança,
é um ponto de inflexão.
Calculando a coordenada y:
Portanto, o ponto de inflexão é
.
Passo 7
f) Assíntota(s), se existirem.
Assíntotas são linhas que a função se aproxima, mas nunca toca. Para polinômios como
, não existem assíntotas horizontais ou verticais, pois elas são definidas para funções racionais onde o denominador pode se anular ou para funções de limite que tendem ao infinito de forma particular. Como nosso polinômio é definido para todo
e não possui divisão por zero ou comportamento assintótico, não existem assíntotas.
Passo 8
g) Construir o gráfico.
Com todas as informações coletadas, podemos esboçar o gráfico da função
:
-
Pontos Críticos:
- Mínimo em
- Máximo em
- Mínimo em
-
Ponto de Inflexão:
-
-
Comportamento nos Intervalos:
- Decrescente para
- Crescente para
- Decrescente para
- Decrescente para
-
Concavidade:
- Concava para cima quando
- Concava para baixo quando
- Concava para cima quando
-
Assíntotas:
- Nenhuma
-
Ponto de Interseção com o Y:
- Quando
: Então, o ponto é .
- Quando
Com essas informações, o gráfico apresenta uma curva que desce até
, onde atinge um mínimo, sobe até
, atingindo um máximo, e depois desce novamente. Há uma inflexão em
, onde a concavidade muda.
Resposta
A função
possui pontos críticos em
(mínimo em
) e
(máximo em
). Ela está decrescente em
e
, e crescente em
. A concavidade é para cima em
e para baixo em
, com ponto de inflexão em
. Não há assíntotas. O gráfico apresenta esses pontos e comportamentos descritos, refletindo as características encontradas.
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 ">
<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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</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=“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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</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=“14.789ex” height=“2.452ex” role=“img” focusable=“false” viewBox=“0 -833.9 6536.6 1083.9”><title>f(x) = x^5 - 5x</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=“1D453” d=“M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 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139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z”></path></g><g data-mml-node=“mo” transform=“translate(1511,0)”><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(2177.8,0)”><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=“msup” transform=“translate(3233.6,0)”><g data-mml-node=“mi”><path data-c=“1D465” d=“M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z”></path></g><g data-mml-node=“TeXAtom” transform=“translate(605,363) scale(0.707)” data-mjx-texclass=“ORD”><g data-mml-node=“mn”><path data-c=“35” d=“M164 157Q164 133 148 117T109 101H102Q148 22 224 22Q294 22 326 82Q345 115 345 210Q345 313 318 349Q292 382 260 382H254Q176 382 136 314Q132 307 129 306T114 304Q97 304 95 310Q93 314 93 485V614Q93 664 98 664Q100 666 102 666Q103 666 123 658T178 642T253 634Q324 634 389 662Q397 666 402 666Q410 666 410 648V635Q328 538 205 538Q174 538 149 544L139 546V374Q158 388 169 396T205 412T256 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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=“14.789ex” height=“2.452ex” role=“img” focusable=“false” viewBox=“0 -833.9 6536.6 1083.9”><title>f(x) = x^5 - 5x</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=“1D453” d=“M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 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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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217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z”></path></g><g data-mml-node=“mo” transform=“translate(1511,0)”><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(2177.8,0)”><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=“mfrac” transform=“translate(3233.6,0)”><g data-mml-node=“msup” transform=“translate(671.8,394) scale(0.707)”><g data-mml-node=“mi”><path data-c=“1D465” d=“M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 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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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