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8Central Asian Journal of
Education and Innovation
ARTICLE INFO
BIRINCHI TARTIBLI HOSILA YORDAMIDA FUNKSIYANING EKSTREMUMGA TEKSHIRISH, FUNKSIYANING EKSTREMUMLARI.
Latipova Shahnoza Salim qizi
Osiyo Xalqaro Universiteti "Umumtexnik fanlar" kafedrasi o'qituvchisi
[email protected] https://doi.org/10.5281/zenodo.10686679
ABSTRACT
Qabul qilindi: 10-February 2024 yil Ma'qullandi: 15- February 2024 yil Nashr qilindi: 21- February 2024 yil
KEY WORDS
Hosila, o'suvchi, interval, qiymat, argument.
Ushbu maqolada funksiyani o'sish va kamayish oraliqlariga tekshirish haqida ma'lumot keltirilgan va misollar ishlab ko'rsatilgan.
Ta'rif: Agar argumentning ^ oraliqqa tegishli katta qiymatiga funksiyaning katta
tengsizlikdan, bunda ^2*1*2 lar (a}b)(a,b)
V S X v [>
qiymati mos kelsa, ya'ni 2 12 intervalga tegishli, №2) > №i)/(*2) > /(* 1) tengsizlik kelib chiqsa.u holda y = -V — f(x) funksiya shu (a,b) intervalda o'suvchi funksiya deyiladi.
Ta'rif: Agar biror (a,b) intervalda argumentning katta qiymatiga funksiyaning kichik qiymati
y n y y ~> v
mos kelsa, ya'ni agar 2 12 - 1 tengsizlikdan, bunda
x^ € (a,b),f(x2)< fOcJx^x2 G (a, b),f(x2) < /(xj tengsizlik kelib chiqsau
holda ~ f№y ~ fOO funksiya (a,b) intervalda kamayuvchi funksiya deyiladi. Teorema.(funksiya o'suvchi bo'lishining zaruriy sharti) Agar (a,b) intervalda
differensiallanuvchi ^ — ^ (-*).) — /00 funksiya o'suvchi bo'lsa, u holda bu funksiyaning hosilasi intervalning hamma nuqtasida manfiy bo'lmasligi zarur, ya'ni barcha X E E ucjlun
/ / (x) - 0
Teorema.(funksiya o'suvchi bo'lishining yetarlilik sharti) Agar [a,b] kesmada uzluksiz
bo'lgan ~ f№y ~ J funksiya har bir ichki nuqtada musbat hosilaga ega bo'lsa,u holda bu funksiya [a,b] kesmada o'suvchi bo'ladi.
Teorema.(funksiya kamayuvchi bo'lishining yetarlilik sharti ) Agar [a,b] kesmada uzluksiz
y ~ ~ /00 funksiya bu kesmaning har bir ichki nuqtasida manfiy hosilaga ega
bo'lsa,u holda bu funksiya [a,b] kesmada kamayuvchi bo'ladi. y = 3x6y = 3x6
l-misol.
funksiyaning monotonlik intervallarini toping.
Yechish. Y' hosilani topamiz: y'=18
X5X:i
Aytaylik f(x) funksiya (a,b) intervalda aniqlangan va хоШ(а;Ь) bo'lsin.
Ta'rif. Agar xo nuqtaning shunday (хо-Ш;хо+Ш) atrofi mavjud bo'lib, shu atrofdan olingan ixtiyoriy x uchun f(x)mf(xo) [/(х)Щхо) ) tenglik o'rinli bo'lsa, u holda xo nuqta f(x) funksiyaning maksimum ( minimum ) nuqtasi, f(xo) esa funksiyaning maksimumi ( minimumi ) deb ataladi.
Ta'rif. Agar xo nuqtaning shunday 1-
chizma
atrofi (хо-Ш;хо+Ш) mavjud bo'lib, shu atrofdan olingan ixtiyoriy хШхо uchun f(x)<f(xo) ( f(x)>f(xo) ) tengsizlik o'rinli bo'lsa, u holda f(x) funksiya xo nuqtada qat'iy maksimumga ( minimumga ) ega deyiladi.
Funksiyaning maksimum va minimum nuqtalari funksiyaning ekstremum nuqtalari, maksimum va minimum qiymatlari funksiyaning ekstremumlari deb ataladi. Shuningdek, f(x) funksiya (a,b) intervalda bir qancha maksimum va minimumlarga ega bo'lishi, maksimum qiymati uning ba'zi bir minimum qiymatidan kichik bo'lishi ham mumkin. Masalan grafigi 1-chizmada ko'rsatilgan y=f(x) funksiya uchun x=a nuqtada lokal maksimum, x=b nuqtada lokal minimum mavjud bo'lib, f(a)<f(b) tengsizlik o'rinli.
Eslatma 1.Agar funksiya [a,b] kesmada aniqlangan bo4sa,bu funksiya o'zining maksimum va minimumlariga x ning shu kesma ichidagi qiymatlaridagina erishadi.
Eslatma 2. Funksiyaning [a,b] kesmadagi maksimum va minimumlari har doim ham uning shu kesmadagi eng katta yoki eng kichik qiymati bo'lavermaydi: maksimum nuqtasida funksiya eng katta qiymatni maksimum nuqtasiga yetarlicha yaqin nuqtalardagi qiymatlariga nisbatangina qabul qiladi.Maksimum minimumdan kichik bo'lib qolishi mumkin
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