Asosiy tarkibga o'tish

4.3-bo'lim : Minimal va maksimal qiymatlar

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

The graph of \(f\left( x \right)\) on \( - 5 \le x \le 9\). Starting at approximately \(\left( { - 5,6} \right)\) the graph decreases to a valley at approximately \(\left( { - 3, - 1} \right)\), rises very slightly to a small peak at approximately \(\left( { - 2, - 0.5} \right)\), then drops to another valley at approximately \(\left( { - 1, - 1} \right)\). It then rises steeply to a peak at approximately \(\left( {1,3} \right)\), drops sharply to a valley at \(\left( {2, - 4} \right)\), rises again to a peak at approximately \(\left( {6,8} \right)\) and then falls steeply, crossing the x-axis near \(x = 8\) and continuing down to approximately \(\left( {9, - 6} \right)\).
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: \(\left( {6,8} \right)\)

Mutlaq minimal: \(\left( {9, - 6} \right)\)

Nisbiy maksimallar: \(\left( {1,3} \right)\) va \(\left( {6,8} \right)\)

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