Ngā Wāhi Tino Rahi o te Uara Whakahoki Iti me te Uara Whakahoki Mōrahi
I roto i te pāngarau me te tātari, he mea nui te ariā o ngā pūwāhi tino nui, i roto i ngā momo mara pūtaiao me ngā mahi o ia rā. Ko ngā pūwāhi tino nui, e pā ana ki ngā pūwāhi i runga i te kauwhata o tētahi mahi e tae atu ai te mahi ki tōna uara iti rawa, mōrahi rānei, he mahi nui tā rātou ki te tautuhi i ngā āhuatanga nui o te mahi. I roto i tēnei tuhinga, ka tūhuratia e mātou te ariā o ngā pūwāhi tino nui, me te aro nui ki ngā uara whakahoki iti rawa me te mōrahi.
Te Whakamāramatanga o te Pūwāhi Tino Rahi
Ko ngā pūwāhi mutunga o tētahi mahi ko ngā pūwāhi e tae atu ai te mahi ki tētahi uara iti rawa, mōrahi rānei. I te nuinga o te wā, ka taea te whakarōpū i ēnei pūwāhi penei:
1. Pūwāhi Mōkito Ā-Rohe: Ko te pūwāhi \( x \) he pūwāhi mōkito ā-rohe o tētahi mahi \( f(x) \) mēnā he wā \( I \) kei roto \( x \) kia rite ai mō ngā \( x \in I \), \( f(x) \ge f(x_0) \).
2. Pūwāhi Mōrahi Ā-Rohe: Ko te pūwāhi \( x \) he pūwāhi mōrahi ā-rohe o te mahi \( f(x) \) mēnā he wā \( I \) kei roto \( x \) kia rite ai mō ngā \( x \in I \), \( f(x) \le f(x_0) \).
Uara Whakahoki Iti rawa me te Uara Whakahoki Mōrahi
Ko te uara whakahoki, te uara mahi rānei o tētahi pūwāhi tino nui, he mea nui ngā mōhiohio i roto i ngā momo tono pūtaiao, hangarau, me te ōhanga. Ko ngā momo uara whakahoki matua e rua ko:
– Uara Whakahoki Mōkito: Ko te uara iti rawa ka tae atu te mahi ki tōna pito mutunga.
– Uara Whakahoki Mōrahi: Ko te uara nui rawa atu ka tae atu te mahi ki tōna pito mutunga.
Te Tātai i ngā Pūwāhi Tino Rahi
I te nuinga o te wā, ko te whakatau i ngā pūwāhi tino nui e uru ana ki ngā tikanga tātaitai rerekētanga. Anei ngā mahi whānui mō te kimi i ngā pūwāhi tino nui o tētahi mahi tonu \( f(x) \):
1. Te Whakarerekētanga o te Mahi: Whakatauhia te pānga tuatahi o te mahi, \( f'(x) \).
2. Ōrite ki te Kore: Kimihia te otinga mō te whārite \( f'(x) = 0 \).
3. Whakamātautau Pūwāhi Tino Pai: Ko ngā pūwāhi kei reira a \( f'(x) = 0 \) he pūwāhi tino pai. Hei whakaū mēnā he pūwāhi tino nui ēnei pūwāhi, me tirotiro tātou i te tuarua o ngā pānga o \( f”(x) \):
– Mena ko te \( f”(x) > 0 \), ko te pūwāhi he pūwāhi iti rawa ā-rohe.
– Mena ko te \( f”(x) < 0 \), ko te pūwāhi he pūwāhi mōrahi ā-rohe.
Hei tauira māmā, me whakaaro tātou ki te mahi tapawhā \( f(x) = x^2 - 4x + 3 \): 1. Te Rerekētanga Mahi: \( f'(x) = 2x - 4 \).
2. He rite ki te Kore: \( 2x - 4 = 0 \Rightarrow x = 2 \).
3. Whakamātautau Pūwāhi Hiranga me te Tuarua o ngā Whakapūtanga: \( f''(x) = 2 \) (he pai tonu).
Nō reira, ko \( x = 2 \) he pūwāhi iti rawa ā-rohe nā te mea he pai te pānga tuarua.
Te Whakamahinga o ngā Pūwāhi Tino Rahi i roto i ngā Marae Ōhanga Rerekē I roto i te ōhanga, ka whakamahia pinepine ngā pūwāhi tino rahi hei whakatau i ngā pūwāhi tino pai i roto i ngā horopaki rerekē, pēnei i te whakaputa me ngā utu. Hei tauira, ka hiahia pea tētahi kamupene ki te whakanui ake i ngā hua, ki te whakaiti rānei i ngā utu whakaputa. He maha ngā wā ka whai āhua ngā pānga painga, ngā pānga utu rānei e āhei ai te whakamahi i te tātaitai ki te kimi i ngā tohu mōrahi, mōkito rānei.
Te Hangarau me te Ahupūngao I roto i te hangarau, ka whakamahia ngā āhuatanga tino nui, i roto i te hoahoa me te tātari i ngā hanganga. Mā te tautuhi i te pūwāhi o te ahotea nui, te rerekētanga rānei, ka āwhina i te karo i te rahunga o te rauemi, me te whakarite hoki i te hoahoa tino pai.
Koiora me te Taiao I roto i te koiora me te taiao, ka whakamahia te ariā o ngā pūwāhi tino nui hei whakatauira i ngā taupori me ngā pūnaha rauropi. Mā te kimi i te pūwāhi mōrahi o tētahi taumahi taupori ka āwhina i te mārama ki ngā āhuatanga e eke ai te taupori ki te kaha mōrahi o tōna taiao.
Ngā Tauira o te Uauatanga me te Rerekētanga Kāore i te ngāwari te kimi i ngā pūwāhi tino nui, inā koa i ngā mahi uaua ake: 1. Ngā Mahi Kore-Rārangi me ngā Mahi Maha-Taurangi: Mō ngā mahi he nui atu i te kotahi te taurangi, pērā i te \( f(x, y) \), ko te tukanga he whakaoti i tētahi pūnaha o ngā whārite pārōnaki ā-wāhanga. Ka nui te hiranga o te whakamahinga o ngā rauropi tau me ngā pūmanawa rorohiko.
2. Ngā Mahi Motukore: I ētahi wā, tera pea he momotukore o ngā mahi e aukati ana i ngā tikanga noa o te wehewehe, e hiahiatia ana ētahi atu huarahi pēnei i te hōtaka raina, te arotautanga tau rānei.
Hei tauira, ko te mahi taurangi-toru \( f(x, y, z) = x^2 + y^2 + z^2 \) he pūwāhi iti rawa kei \( (0, 0, 0) \) engari ko te arotau i ngā mahi kāore i te whai i tētahi āhua paerewa pēnei ka uaua pea, ā, me whai tikanga tau auau.
Te Arotautanga Tau I roto i te nuinga o ngā tono tūturu, kāore i te rawaka ngā tikanga tātari i ngā wā katoa nā te uaua o ngā mahi e tātarihia ana. He maha ngā wā ka whakamahia ngā tikanga arotau tau pēnei i te raupapatanga hekenga gradient, te tikanga simplex, te raupapatanga ira rānei. Ka whakamahia e tēnei tikanga ngā tāruarua e anga ana i te otinga ki ngā pūwāhi tino nui e ai ki ngā ture kua whakaritea i mua.
Whakamutunga He mea nui te mārama ki ngā pito rawa o tētahi mahi i roto i ngā momo mara pūtaiao me ngā tono mahi. Mā roto i ngā tikanga tātaitai, ka taea e tātou te tautuhi i ngā uara whakahoki iti rawa me te mōrahi o te mahi, e whakarato ana i ngā māramatanga nui mō te whakatau kaupapa me te whakaoti rapanga. He maha ngā wā ka honoa ngā tikanga tātari me ngā taputapu arotau tau hei whakahaere i ngā mahi uaua ake, hei kimi hoki i ngā otinga tino pai i roto i ngā horopaki rerekē. Nā reira, kei te whai tikanga tonu te ariā o ngā pito rawa, ā, he whānui te whakamahinga i roto i te pūtaiao me te hangarau hou.
