Asosiy tarkibga o'tish

4.3-bo'lim : Minimal va maksimal qiymatlar

1. Quyida \(f\left( x \right)\) funksiyasining grafigi keltirilgan. Funksiyaning barcha nisbiy ekstremal va mutlaq ekstremallarini aniqlang.

The graph of \(f\left( x \right)\) on \( - 4 \le x \le 5\). Starting at approximately \(\left( { - 4, - 1} \right)\) the graph decreases to a valley at approximately \(\left( { - 3, - 1.5} \right)\), then increases to a peak at \(\left( { - 1,2} \right)\), then falls steadily to a sharp valley at \(\left( {2, - 6} \right)\), then rises steeply to a peak at \(\left( {4,5} \right)\) and finally decreases a little to approximately \(\left( {5,2} \right)\) at the right end.
Yechim

Bu muammo uchun haqiqatan ham unchalik ko'p narsa yo'q. Biz bilamizki, mutlaq ekstremallar grafikdagi eng yuqori/eng past nuqtadir va ular grafikning oxirgi nuqtalarida yoki ichki qismida paydo bo'lishi mumkin. Boshqa tomondan, nisbiy ekstremallar grafikdagi "to'qnashuvlar" yoki "to'qnashuvlar" bo'lib, bu nuqta atrofidagi mintaqada "zarba" maksimal yoki minimal bo'ladi. Shuni ham yodda tutingki, nisbiy ekstremallar oraliqning oxirgi nuqtalarida emas, faqat grafikning ichki qismida paydo bo'ladi.

Shuni ham eslatib o'tamizki, nisbiy ekstremal ham mutlaq ekstremal bo'lishi mumkin.

Shunday qilib, bizda quyidagi mutlaq/nisbiy ekstrema bor.

Mutlaq maksimal: \(\left( {4,5} \right)\)

Mutlaq minimal: \(\left( {2, - 6} \right)\)

Nisbiy maksimallar: \(\left( { - 1,2} \right)\) va \(\left( {4,5} \right)\)

Nisbiy minimallar: \(\left( { - 3, - 2} \right)\) va \(\left( {2, - 6} \right)\)