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Cell["Ekstreemum\[UDoubleDot]lesanded", "Section",
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Cell[TextData[StyleBox["M\[OTilde]isted",
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 StyleBox["Ekstreemum",
  FontWeight->"Bold"],
 " (algallikas on ladina s\[OTilde]na ",
 StyleBox["extremum",
  FontSlant->"Italic"],
 " \[OpenCurlyQuote]\[ADoubleDot]\[ADoubleDot]r, piir\[CloseCurlyQuote]) ehk ",
 StyleBox["ekstremaalv\[ADoubleDot]\[ADoubleDot]rtus",
  FontSlant->"Italic"],
 " on funktsiooni maksimum v\[OTilde]i miinimum. ",
 StyleBox["Ekstreemumkoht",
  FontWeight->"Bold"],
 " - \[UDoubleDot]he muutuja funktsiooni argumendi niisugune v\[ADoubleDot]\
\[ADoubleDot]rtus, mille korral see funktsioon saavutab oma ekstreemumi. "
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Cell[TextData[{
 StyleBox["Ekstreemumpunkt:",
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 "  \[LineSeparator]1. \[UDoubleDot]he muutuja funktsiooni  ekstreemumkohale \
vastav punkt selle funktsiooni graafikul.\n2. \
m\[ADoubleDot]\[ADoubleDot]ramispiirkonna punkt, milles vaadeldav mitme \
muutujaga funktsioon saavutab oma ekstreemumi."
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 "Funktsioonil v\[OTilde]ivad olla ekstreemumid ainult ",
 StyleBox["kriitilistes punktides",
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 ". Mis on kriitiline punkt?"
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 "Punkti ",
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 " nimetatakse funktsiooni ",
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 "(",
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 ") ",
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 ", kui ",
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  FontSlant->"Italic"],
 "(",
 StyleBox["a",
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 ") = 0 (statsionaarne punkt)\nv\[OTilde]i kui kohal ",
 StyleBox["a",
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 " ei ole sel funktsioonil tuletist."
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Cell[TextData[{
 StyleBox["Kokkuv\[OTilde]te:",
  FontWeight->"Bold"],
 " Ekstreemumpunkt on kriitiline punkt, milles funktsiooni tuletis muudab m\
\[ADoubleDot]rki."
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Cell["\<\
Kui tuletise m\[ADoubleDot]rk muutub argumendi \
v\[ADoubleDot]\[ADoubleDot]rtuse kasvades plussist miinuseks, siis on \
tegemist maksimumiga.
Kui tuletise m\[ADoubleDot]rk muutub argumendi \
v\[ADoubleDot]\[ADoubleDot]rtuse kasvades miinusest plussiks, siis on \
tegemist miinimumiga.\
\>", "Text",
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Cell[TextData[{
 "Statsionaarsete punktide (",
 StyleBox["f ' ",
  FontSlant->"Italic"],
 "(",
 StyleBox["a",
  FontSlant->"Italic"],
 ") = 0) puhul aitab maksimume-miinimume eristada funktsiooni teine tuletis. "
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 "Kui ",
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Cell[CellGroupData[{

Cell[TextData[StyleBox["N\[ADoubleDot]ited",
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Cell[TextData[{
 StyleBox["N\[ADoubleDot]ide 1.",
  FontWeight->"Bold"],
 " Ristk\[UDoubleDot]likukujulisest papit\[UDoubleDot]kist, mille m\[OTilde]\
\[OTilde]tmed on 50 cm ja 80 cm, tuleb valmistada kaaneta karp. Selleks l\
\[OTilde]igatakse papit\[UDoubleDot]ki nurkadest \[ADoubleDot]ra \
v\[OTilde]rdsed ruudud ja murtakse servad \[UDoubleDot]les. Kui suur peab \
olema v\[ADoubleDot]ljal\[OTilde]igatavate ruutude k\[UDoubleDot]lg, et karbi \
ruumala oleks suurim?"
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Cell[CellGroupData[{

Cell["\<\
Olgu v\[ADoubleDot]ljal\[OTilde]igatavate ruutude k\[UDoubleDot]lje pikkus x \
cm. 
On ilmne, et 0 < x < 25.\
\>", "Text",
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Cell[CellGroupData[{

Cell["\<\
Papit\[UDoubleDot]kist valmistatava karbi p\[OTilde]hjaks on ristk\
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Seega karbi p\[OTilde]hja pindala on\
\>", "Text",
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Cell["(50 - 2 x) (80 - 2 x) ", "Text",
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Cell["ja ruumala on x\[CenterDot](50 - 2 x) (80 - 2 x).", "Text",
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Cell["\<\
Et karbi ruumala V s\[OTilde]ltub \[ADoubleDot]ral\[OTilde]igatavate ruutude \
k\[UDoubleDot]lje pikkusest x, siis v\[OTilde]ib ruumala vaadelda muutuja x \
funktsioonina: \
\>", "Text",
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Cell[TextData[{
 "Leiame funktsiooni V (x) =  4",
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 " + 4000x maksimumkoha vahemikust ]0; 25[."
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Cell[TextData[{
 "V' (x) = 12 ",
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 "- 520 x + 4000 = 0 \[DoubleRightArrow] ",
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 " = 10 v\[OTilde]i ",
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 " = ",
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   FractionBox["100", "3"], TraditionalForm]]],
 "."
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Cell[TextData[{
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 " > 25, seet\[OTilde]ttu uurime seda, kas tegemist on maksimumi v\[OTilde]i \
miinimumiga ainult\nkohal x = 10. "
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Cell["Kasutame teist tuletist:", "Text",
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Cell["V''(x) = 24x - 520;   V''(10) = 240 - 520 = -280 < 0.", "Text",
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Cell["Seega kohal 10 on funktsiooni V(x) maksimumkoht.", "Text",
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Cell["\[CapitalADoubleDot]ral\[OTilde]igatava ruudu k\[UDoubleDot]lg peab \
olema 10 cm.", "Text",
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Cell[TextData[{
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 " Silindrikujulise konservipurgi ruumala on V. Millised peavad olema \
konservipurgi m\[OTilde]\[OTilde]tmed, et purgi valmistamiseks kuluks v\
\[OTilde]imalikult v\[ADoubleDot]he plekki (valtsimist ei arvestata!)?"
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Siit j\[ADoubleDot]reldub, et konservipurgi p\[OTilde]hja l\[ADoubleDot]bim\
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 " Keskp\[ADoubleDot]eval on purjekas aurikust 20 km kaugusel l\[OTilde]unas. \
Purjekas liigub kiirusega 20 km/h itta, aurik kiirusega 40 km/h \
l\[OTilde]unasse. Kas aurikult on n\[ADoubleDot]ha purjekat, kui n\
\[ADoubleDot]htavus on 10 km?"
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 "Lepime kokku, et aega ",
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 " m\[OTilde]\[OTilde]dame tundides, kiiruse \[UDoubleDot]hik olgu km/h.\n\
Keskp\[ADoubleDot]eval olgu ",
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 "Ajahetkel ",
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 " \[GreaterSlantEqual] 0 on purjekas liikunud 20",
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 " km itta ja aurik 40",
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 "Siit v\[OTilde]ime j\[ADoubleDot]reldada, et aurikult v\[OTilde]ib n\
\[ADoubleDot]ha purjekat. K\[OTilde]ige parem peaks n\[ADoubleDot]htavus \
olema\n",
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 " Marjuline on metsa sees 3 km kaugusel sirgest kiirteest. Kiirteel, 6 km \
ida pool on bussipeatus. Marjuline liigub metsas kiirusega 4 km/h, m\
\[ODoubleDot]\[ODoubleDot]da asfalti kiirusega 5 km/h. Millisesse punkti tee \
\[ADoubleDot]\[ADoubleDot]res peab marjulk\[ADoubleDot]ija suunduma, et j\
\[OTilde]uda bussipeatusse minimaalse ajaga?"
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Cell[TextData[StyleBox["Kahe muutujaga funktsiooni ekstreemumid",
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Cell["\<\
Kahe muutuja funktsioon on seos, mis seab j\[ADoubleDot]rjestatud arvupaarile \
(x; y) vastavusse
\[UDoubleDot]he ja ainult \[UDoubleDot]he reaalarvu z = f (x, y). \
\>", "Text",
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Cell["\<\
Kahe muutuja funktsiooni z = f (x, y)  graafikuks on pind 3-m\[OTilde]\
\[OTilde]tmelises ruumis. 
Argumendipaar (x, y) on punkt xy -tasandil ja funktsiooni v\[ADoubleDot]\
\[ADoubleDot]rtus z \"pinna k\[OTilde]rgus\" selles punktis. Eristada tuleb \
ekstreemumpunkte ja sadulpunkte.\
\>", "Text",
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Millise temperatuuri ja niiskusetaseme juures kasvavad need kasvuhoonetaimed \
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Kui suur on see kasvamiskiirus?\
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Ekstreemumite olemasolu kahtlustame punktides, kus need osatuletised on v\
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kujundatud suurima t\[ADoubleDot]ispindalaga silinder. Arvuta silindri p\
\[OTilde]hja raadius. (5 cm). Kuidas lahendada seda \[UDoubleDot]lesannet, \
kui koonuse raadius on r ja k\[OTilde]rgus h."
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 " Koonusekukujulisest puut\[UDoubleDot]vest, mis on 24 m pikkune ja mille l\
\[ADoubleDot]bim\[OTilde]\[OTilde]t juure kohalt on 0,48 m tahetakse raiuda \
maksimaalse ruumalaga silindrikujuline palk. Kui pikk t\[UDoubleDot]kk tuleb \
ladvapoolsest otsast maha saagida?"
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 " Koonuse k\[OTilde]rgus on v\[OTilde]rdne diameetriga. Koonuse sisse on \
kujundatud maksimaalse ruumalaga silinder. Leia silindri ja koonuse ruumalade \
suhe."
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 "Pilt mille k\[OTilde]rgus on 1,4 m, ripub seinal nii, et ta alumine \
\[ADoubleDot]\[ADoubleDot]r on 1,8 m k\[OTilde]rgemal vaatleja silmadest. Kui \
kaugel peab seisma vaatleja, et pilt oleks k\[OTilde]ige paremini n\
\[ADoubleDot]htav (vaatenurk oleks maksimaalne)? (2,4 m)"
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\[ADoubleDot]r aga ",
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 " meetrit k\[OTilde]rgemal, vaatleja silma optimaalne kaugus seinast olgu x, \
pildi alumine \[ADoubleDot]\[ADoubleDot]r  paistku kaugusel ",
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\[ADoubleDot]r aga nurga ",
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Kui keegi tahab seda \[UDoubleDot]lesannet arvulisel kujul lahendada (muide \
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siis olgu antud punktide A ja B koordinaadid, n\[ADoubleDot]iteks A(0; 6), \
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 " Kauboi on koos oma hobusega laagris punktis A. Ta peab hobust j\[OTilde]es \
(j\[OTilde]e kaldal) jootma ja \nmetsas (metsa servas) s\[ODoubleDot]\
\[ODoubleDot]tma ning tulema seej\[ADoubleDot]rel tagasi punktis A olevasse \
laagrisse, tehes \nseda nii, et l\[ADoubleDot]bitud vahemaa oleks v\
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Seega, peab ta leidma j\[OTilde]e kaldal punkti B ja metsaservas punkti C nii, 
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Punkti A asukohta nurga sees peab saama vabalt valida; seda, mis 
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