Extremes ntawm kev khiav dej num - cov lus yooj yim txog lub complex

Yuav kom to taub dab tsi yog qhov point ntawm extremum ntawm ib tug muaj nuj nqi tsis tau paub txog lub xub ntiag ntawm tus thawj thiab thib ob derivative thiab to taub txog lawv lub cev lub ntsiab lus. Ua ntej koj yuav tsum to taub raws li nram no:

• extrema ntawm cov nuj nqi yog maximized, los yog, conversely, txo tus nqi ntawm cov nuj nqi nyob rau hauv ib tug arbitrarily me me zej zog;
• nyob rau extremum yuav tsum tsis muaj kis muaj nuj nqi.

Thiab tam sim no rau ib yam khoom, tsuas nyob rau hauv cov lus yooj yim. Saib nyob rau ntxeev ntawm ib tug cwj mem. Yog hais tias tus kov positioned mas sau ntawv kawg upward, ces feem ntau ntawm cov pob yuav nruab nrab extremum - lub siab tshaj plaws taw tes. Nyob rau hauv cov ntaub ntawv no peb hais txog lub siab tshaj plaws. Tam sim no, Yog hais tias koj tig lub sau ntawv xaus cia, ces tus pob yuav tsum muaj tsawg kawg seredke twb functions. Siv cov duab muab ntawm no, uas teev tseg tej zaum yuav tam sim no rau manipulation sau mem hluav. Yog li ntawd extrema ntawm cov nuj nqi - nws yog ib txwm ib tug tseem ceeb point: nws highs los yog lows. Cov uas nyob ib sab ib feem ntawm daim duab yuav ua tau arbitrarily ntse los yog tus, tab sis nws yuav tsum muaj nyob rau ntawm ob sab, tab sis nyob rau hauv cov ntaub ntawv no, tus taw tes yog lub ncov. Yog hais tias daim ntawv yog tam sim no nyob rau hauv tsuas yog ib qho sab, lub taw tes rau ntawm no extremum yuav tsis tau, txawm hais tias nyob rau ib sab ntawm lub extremum tej yam kev mob uas tau ntsib. Tam sim no peb kawm txog tej extremes ntawm kev khiav dej num ntawm ib tug scientific point of view. Yog li ntawd hais tias tus taw tes, kuj yuav tau ib tug extremum, nws yog tsim nyog thiab txaus tias:

• thawj derivative yog sib npaug zos rau zero los yog tsis muaj nyob rau hauv lub taw tes;
• thawj derivative hloov kos npe rau ntawm no point.

Tej yam kev mob kho dog dig txawv nyob rau hauv cov nqe lus ntawm derivatives ntawm ntau dua-kev txiav txim muaj nuj nqi uas yog differentiable ntawm tus taw tes nws yog txaus uas yuav tau ib tug khib-thiaj derivative, tsis zoo tib mus pes tsawg txawm lub fact tias tag nrho cov derivatives ntawm ib tug qis kev txiav txim thiab yuav tsum muaj pes tsawg. Qhov no yog feem ntau yooj yim los txhais lus rau theorems los ntawm cov phau ntawv ntawm kev kawm leb siab. Tab sis nws yog tsim nyog los ntxiv no point raws li ib qho piv txwv rau neeg zej zog. Lub hauv paus yog ib tug dog dig parabola. Nyuam nyob rau xoom taw tes nws muaj ib tug yam tsawg kawg nkaus. Heev me ntsis kev kawm txog zauv:

• thawj derivative ntawm (X ^{2) |} = 2 x, 2 x rau lub voj point = 0;
• tus thib ob derivative (2 x) ^| = 2, rau lub voj point 2 = 2.

Tej yam yooj yim yam piv txwv tej yam kev mob txiav txim extrema ntawm cov nuj nqi rau tus thawj txiav txim thiab kev ua derivatives. Koj muaj peev xwm ntxiv mus rau qhov tias tus thib ob derivative yog cia li heev derivative ntawm khib kev txiav txim, tsis zoo tib mus pes tsawg, uas hais saum toj no cia. Thaum nws los txog rau hauv lub extremes ntawm ib tug muaj nuj nqi ntawm ob qhob, lub tej yam kev mob yuav tsum tau ntsib rau ob nqe lus. Thaum muaj ib tug generalization, ces nyob rau hauv lub chav kawm yog cov ib nrab derivatives. Uas yog tsim nyog rau lub neej ntawm ib tug extremum ntawm tus taw tes hais tias ob thawj derivatives yog pes tsawg, los yog tsawg kawg yog ib tug ntawm lawv tsis ua ib ke. Rau sufficiency muaj extremum soj ntsuam qhia sawv cev rau cov khoom ntawm qhov txawv ntawm lub thib ob kev txiav txim thiab lub square ntawm lub mixed thib ob-thiaj derivative muaj nuj nqi. Yog hais tias qhov no qhia yog ntau tshaj pes tsawg, ces cov extremum tshwm sim, thiab yog hais tias muaj yog sib npaug zos rau zero, ces cov nqe lus nug tseem qhib, thiab qhov yuav tsum tau rau kev ntxiv kev tshawb fawb.