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4.6-bo'lim : Grafik shakli, II qism

7. \(R\left( z \right) = z{\left( {z + 4} \right)^{\,{\frac{2}{3}}}}\) uchun quyidagi savollarning har biriga javob bering.

  1. Funktsiya uchun mumkin bo'lgan burilish nuqtalari ro'yxatini aniqlang.
  2. Funktsiya yuqoriga va pastga botiq bo'lgan intervallarni aniqlang.
  3. Funksiyaning burilish nuqtalarini aniqlang.
a Funktsiya uchun mumkin bo'lgan burilish nuqtalari ro'yxatini aniqlang. Yechim

Funktsiyaning mumkin bo'lgan burilish nuqtalari ro'yxatini olish uchun bizga 2 kerak bo'ladind funktsiyaning hosilasi, shuning uchun bu erda.

\[\begin{align*}R'\left( z \right) & = {\left( {z + 4} \right)^{\,{\frac{2}{3}}}} + z\left( {{\frac{2}{3}}} \right){\left( {z + 4} \right)^{\, - \,\,{\frac{1}{3}}}} = \frac{{5z + 12}}{{3{{\left( {z + 4} \right)}^{{\frac{1}{3}}}}}}\\ R''\left( z \right) & = \frac{{5\left( {3{{\left( {z + 4} \right)}^{{\frac{1}{3}}}}} \right) - \left( {5z + 12} \right){{\left( {z + 4} \right)}^{ - {\frac{2}{3}}}}}}{{{{\left[ {3{{\left( {z + 4} \right)}^{{\frac{1}{3}}}}} \right]}^2}}} = \frac{{\left[ {15\left( {z + 4} \right) - \left( {5z + 12} \right)} \right]{{\left( {z + 4} \right)}^{ - {\frac{2}{3}}}}}}{{9{{\left( {z + 4} \right)}^{{\frac{2}{3}}}}}} = \underline {\frac{{10z + 48}}{{9{{\left( {z + 4} \right)}^{{\frac{4}{3}}}}}}} \end{align*}\]

E'tibor bering, biz keyingi bosqichda yordam berish uchun har bir qadamda lotinlarni soddalashtirdik. Buni texnik jihatdan qilishingiz shart emas, lekin 2-ga ega bo'lishingiz keraknd lotin "eng oddiy" shaklida, albatta, bu qismga javob olishda yordam beradi.

Endi eslang, mumkin bo'lgan burilish nuqtalari 2nd hosila yo mavjud emas yoki nolga teng. Chunki biz 2 ni soddalashtirdiknd iloji boricha hosila 2 ekanligi aniqnd hosila \(z = - 4\) da mavjud bo'lmaydi (va funktsiya bu nuqtada ham mavjud!). Bundan tashqari, aniq bo'lishi kerakki, 2nd hosilasi \(z = - \frac{48}{10} = - \frac{24}{5}\) da nolga teng.

Ushbu funktsiyaning mumkin bo'lgan kritik nuqtalari quyidagilardir:

\[\underline {z = - \frac{24}{5} = - 4.8\,\,\,\,\,\,\,\,\& \,\,\,\,\,\,\,\,z = - 4} \]

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

Bu qismda ko'p narsa yo'q. Bu erda bizga kerak bo'lgan yagona narsa 2 uchun raqam chizig'ind hosila. Mana bu,

A number line for the second derivative of \(R\left( z \right) = z{\left( {z + 4} \right)^{\frac{2}{3}}}\) with dashed vertical lines at \(z = - \frac{{24}}{5}\) and \(z = - 4\) dividing it into three regions. In the left region \(R''\left( { - 5} \right) = - \frac{2}{9}\) so \(R''\left( z \right) < 0\), in the middle region \(R''\left( { - \frac{9}{2}} \right) = 0.84\) so \(R''\left( z \right) > 0\), and in the right region \(R''\left( { - 3} \right) = 2\) so \(R''\left( z \right) > 0\).

Bundan biz funktsiya uchun quyidagi konkav yuqori/past ma'lumotni olamiz.

\[\require{bbox} \bbox[2pt,border:1px solid black]{{\,\,\,\,\,\,\,{\mbox{Concave Up : }}\left( { - \frac{24}{5}, - 4} \right)\,\,\,\,\,\& \,\,\,\,\,\left( { - 4,\infty } \right)\,\,\,\,\,\,\,\,\,\,\,\,\,{\mbox{Concave Down : }}\left( { - \infty , - \frac{24}{5}} \right)}}\]

c Funksiyaning burilish nuqtalarini aniqlang. Yechim

Ushbu qism uchun biz qilishimiz kerak bo'lgan narsa avvalgi bosqich natijalarini sharhlashdir. Esingizda bo'lsin, egilish nuqtalari konkavlik o'zgargan nuqtalardir (oddiy 2 nuqtadan farqli o'laroq).nd hosila nolga teng yoki mavjud emas). Shuning uchun, bu funksiya uchun yagona burilish nuqtasi,

\[\require{bbox} \bbox[2pt,border:1px solid black]{{z = - \frac{24}{5}}}\]