Asosiy tarkibga o'tish

4.6-bo'lim : Grafik shakli, II qism

13. \(f\left( x \right) = {x^{\,{\frac{4}{3}}}}\left( {x - 2} \right)\) uchun quyidagi savollarning har biriga javob bering.

  1. Funktsiyaning kritik nuqtalarini aniqlang.
  2. Funksiyaning ortishi va kamayishi oraliqlarini aniqlang.
  3. Kritik nuqtalarni nisbiy maksimal, nisbiy minimum yoki hech biri sifatida tasniflang.
  4. Funktsiya yuqoriga va pastga botiq bo'lgan intervallarni aniqlang.
  5. Funksiyaning burilish nuqtalarini aniqlang.
  6. Bosqichlardagi ma'lumotlardan foydalaning (a) – (e) funksiyaning grafigini chizish uchun.
a Funktsiyaning kritik nuqtalarini aniqlang. Yechim

Ushbu muammoning qismlari (oxirgi qism bundan mustasno) xuddi oldingi bo'limdagi asosiy oshirish/kamayish muammolari va ushbu bo'limning oldingi qismidagi asosiy konkavlik muammolari kabi. Shuning uchun biz bu erda juda ko'p tafsilotlarni keltirmaymiz. Agar siz hali ham ushbu muammoning qismlarini qanday ishlashni bilmasangiz, davom etishdan oldin orqaga qaytib, oldingi va/yoki ushbu bo'limning oldingi qismidagi bir nechta asosiy muammolarni hal qilishingiz kerak.

Bizga 1 kerak bo'ladist ishlarni boshlash uchun hosila.

\[f\left( x \right) = {x^{\,{\frac{7}{3}}}} - 2{x^{\,{\frac{4}{3}}}}\hspace{0.5in}\,\,\,\, \to \hspace{0.5in}\,\,\underline {f'\left( x \right) = {\frac{7}{3}}{x^{\,{\frac{4}{3}}}} - {\frac{8}{3}}{x^{\,{\frac{1}{3}}}} = {\frac{1}{3}}{x^{\,{\frac{1}{3}}}}\left( {7x - 8} \right)} \]

E'tibor bering, \({x^{{\frac{1}{3}}}}\)-ni ko'paytuvchilarga ajratishga ajratish orqali biz muhim nuqtalarni tezda ko'rishni biroz osonlashtirdik,

\[\underline {x = 0\,\,\,\,\,\,\,\& \,\,\,\,\,\,x = {\frac{8}{7}} = 1.1429} \]

b Funksiyaning ortishi va kamayishi oraliqlarini aniqlang. Yechim

Bu qismga javob berish uchun bizga kerak bo'lgan yagona narsa 1 uchun raqam chizig'ist hosila.

A number line for the first derivative of \(f\left( x \right) = {x^{\frac{4}{3}}}\left( {x - 2} \right)\) with dashed vertical lines at \(x = 0\) and \(x = \frac{8}{7}\) dividing it into three regions. In the left region \(f'\left( { - 1} \right) = 5\) so \(f'\left( x \right) > 0\), in the middle region \(f'\left( 1 \right) = - \frac{1}{3}\) so \(f'\left( x \right) < 0\), and in the right region \(f'\left( 2 \right) = 2.52\) so \(f'\left( x \right) > 0\).

Bundan biz funktsiya uchun quyidagi oshirish/kamayish ma'lumotlarini olamiz.

\[\require{bbox} \bbox[2pt,border:1px solid black]{{{\mbox{Increasing : }}\left( { - \infty ,0} \right)\,\,\,\,\,\& \,\,\,\,\,\left( {{\frac{8}{7}},\infty } \right)\hspace{0.25in}{\mbox{Decreasing : }}\left( {0,{\frac{8}{7}}} \right)}}\]

c Kritik nuqtalarni nisbiy maksimal, nisbiy minimum yoki hech biri sifatida tasniflang. Yechim

Oldingi bosqichdagi raqamlar chizig'idan biz kritik nuqtalarning quyidagi tasniflarini olamiz.

\[\require{bbox} \bbox[2pt,border:1px solid black]{{x = 0:\,\,\,{\mbox{Relative Maximum}}\hspace{0.5in}\hspace{0.5in}\hspace{0.5in}\,\,\,x = {\frac{8}{7}}:\,\,\,{\mbox{Relative Minimum}}\,}}\]

d Funktsiya yuqoriga va pastga botiq bo'lgan intervallarni aniqlang. Yechim

Bizga 2 kerak bo'ladind mumkin bo'lgan burilish nuqtalari ro'yxatini topish uchun hosila.

\[\underline {f''\left( x \right) = {\textstyle{{28} \over 9}}{x^{\,{\frac{1}{3}}}} - {\frac{8}{9}}{x^{ - \,{\frac{2}{3}}}} = \frac{{28x - 8}}{{9{x^{{\frac{2}{3}}}}}}} \]

Ushbu funktsiya uchun mumkin bo'lgan burilish nuqtalari:

\[\underline {x = 0\hspace{0.5in}\& \hspace{0.5in}x = {\frac{2}{7}} = 0.2857} \]

Konkavlik oraliqlarini olish uchun bizga 2 uchun raqam chizig'i kerak bo'ladind hosila.

A number line for the second derivative of \(f\left( x \right) = {x^{\frac{4}{3}}}\left( {x - 2} \right)\) with dashed vertical lines at \(x = 0\) and \(x = \frac{2}{7}\) dividing it into three regions. In the left region \(f''\left( { - 1} \right) = - 4\) so \(f''\left( x \right) < 0\), in the middle region, whose computation is boxed and joined to its region by an arrow, \(f''\left( {\frac{1}{{10}}} \right) = - 2.7\) so \(f''\left( x \right) < 0\), and in the right region \(f''\left( 1 \right) = \frac{{20}}{9}\) so \(f''\left( x \right) > 0\).

Bundan biz funktsiya uchun quyidagi konkavlik ma'lumotlarini olamiz.

\[\require{bbox} \bbox[2pt,border:1px solid black]{{{\mbox{Concave Up : }}\left( {{\frac{2}{7}},\infty } \right)\hspace{0.5in}{\mbox{Concave Down : }}\left( { - \infty ,0} \right)\,\,\,\,\,\,\& \,\,\,\,\,\,\left( {0,{\frac{2}{7}}} \right)}}\]

e Funksiyaning burilish nuqtalarini aniqlang. Yechim

Oldingi bosqichdagi konkavlik ma'lumotlaridan biz funktsiyaning yagona burilish nuqtasi ekanligini ko'rishimiz mumkin,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{x = {\frac{2}{7}}}}\]

f Bosqichlardagi ma'lumotlardan foydalaning (a) – (e) funksiyaning grafigini chizish uchun. Yechim

Yuqoridagi ma'lumotlardan foydalangan holda ushbu funktsiyaning grafigining eskizi. Ushbu bo'limdagi masalalarda qilganimizdek, biz grafikda chapdan boshlashimiz va o'ng tomonga o'tishimiz mumkin. Buni amalga oshirayotganda, biz birinchi navbatda ortib borayotgan/kamayuvchi ma'lumotlarga e'tibor beramiz va keyin egri chiziqni chizganimizda to'g'ri konkavlikka ega ekanligiga ishonch hosil qilamiz.

A sketch of the graph of \(f\left( x \right) = {x^{\frac{4}{3}}}\left( {x - 2} \right)\). The graph rises steeply from the lower left to a sharp peak at the origin, then falls through the inflection point \(\left( {\frac{2}{7}, - 0.3226} \right)\) to a valley at \(\left( {\frac{8}{7}, - 1.0242} \right)\) before rising again to the upper right. The labeled points are marked on the graph.

E'tibor bering, biz eskizni yaratish uchun kompyuterdan foydalanganimiz sababli, eskizingiz bir xil bo'lmasligi mumkin. Biroq, u yuqoridagi grafikda sanab o'tilgan bir xil nuqtalarga, bir xil asosiy o'sish/kamayuvchi tabiatga va bir xil asosiy konkavlikka ega bo'lishi kerak.