Asosiy tarkibga o'tish

4.5-bo'lim : Grafik shakli, I qism

12. \(g\left( x \right) = x - 2\ln \left( {1 + {x^2}} \right)\) 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.
a Funktsiyaning kritik nuqtalarini aniqlang. Yechim

Bizga 1 kerakst kritik nuqtalarni olish uchun lotin, shuning uchun bu erda.

\[g'\left( x \right) = 1 - 2\frac{{2x}}{{1 + {x^2}}} = \frac{{1 - 4x + {x^2}}}{{1 + {x^2}}}\]

Endi eslang, tanqidiy nuqtalar lotin mavjud bo'lmagan yoki nolga teng bo'lgan joydir. Biz hosilani iloji boricha soddalashtirganimiz va faktorlarga ajratganimiz uchun hosila hamma joyda mavjud bo'lishini aniq ko'rishimiz mumkin (yoki hech bo'lmaganda har qanday haqiqiy sonlar uchun maxraj nolga teng bo'lmaydi ...). Surat va shuning uchun hosila nolga teng ekanligini aniqlash uchun bizga kvadrat formula ham kerak bo'ladi. Bu funktsiyaning kritik nuqtalari quyidagilardan iborat:

\[\require{bbox} \bbox[2pt,border:1px solid black]{{x = 2 \pm \sqrt 3 = 0.2679,\,\,\,3.7321}}\]

b Funksiyaning ortishi va kamayishi oraliqlarini aniqlang. Yechim

Funktsiyaning o'sishi/kamayishi ma'lumotlarini aniqlash uchun bizga kerak bo'lgan narsa - hosila uchun tezkor raqam chizig'i. Mana raqam chizig'i.

A number line for the derivative of \(g\left( x \right) = x - 2\ln \left( {1 + {x^2}} \right)\) with dashed vertical lines at \(x = 0.2679\) and \(x = 3.7321\) dividing it into three regions. In the left region \(g'\left( 0 \right) = 1\) so \(g'\left( x \right) > 0\), in the middle region \(g'\left( 1 \right) = - 1\) so \(g'\left( x \right) < 0\), and in the right region \(g'\left( 4 \right) = 0.06\) so \(g'\left( x \right) > 0\).

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

\[\require{bbox} \bbox[2pt,border:1px solid black]{{{\mbox{Increasing : }}\left( { - \infty ,0.2679} \right)\,\,\,\,\,\& \,\,\,\,\,\left( {3.7321,\infty } \right)\hspace{0.25in}\hspace{0.25in}{\mbox{Decreasing : }}\,\,\left( {0.2679,3.7321} \right)}}\]

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

Oldingi bosqichdagi ma'lumotni oshirish/kamaytirish bilan biz 1-dan foydalanib, muhim nuqtalarni osongina tasniflashimiz mumkinst hosilaviy test. Bu erda kritik nuqtalar funktsiyalarining tasnifi.

\[\require{bbox} \bbox[2pt,border:1px solid black]{\begin{align*}x & = 0.2679\,\,\,\,\,\,:\,{\mbox{ Relative Maximum}}\\ x & = 3.7321\,\,\,\,\,\,\,:\,{\mbox{ Relative Minimum}}\end{align*}}\]