Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada$$\left(2 - 2 x\right) e^{- x^{2} + 2 x} = 0$$
Resolvermos esta ecuaciónRaíces de esta ecuación
$$x_{1} = 86.3691593312175$$
$$x_{2} = 14.9992809268135$$
$$x_{3} = 78.3814824148947$$
$$x_{4} = 28.6208538623602$$
$$x_{5} = 96.3566621914633$$
$$x_{6} = 58.4273097814622$$
$$x_{7} = 34.554507149414$$
$$x_{8} = -92.1055171244878$$
$$x_{9} = 13.125084358924$$
$$x_{10} = -14.6430797877549$$
$$x_{11} = -72.1343857669297$$
$$x_{12} = 48.4647034274775$$
$$x_{13} = 20.7719971377255$$
$$x_{14} = 54.4405897196054$$
$$x_{15} = 94.358947085372$$
$$x_{16} = -40.2388692669542$$
$$x_{17} = -26.3615719551025$$
$$x_{18} = 6.66015534698603$$
$$x_{19} = 22.7238244881469$$
$$x_{20} = 72.392536498399$$
$$x_{21} = -58.1662052407992$$
$$x_{22} = -78.1241933853779$$
$$x_{23} = -76.1274147285399$$
$$x_{24} = 84.372018487179$$
$$x_{25} = -24.3901348160467$$
$$x_{26} = 100.255036987721$$
$$x_{27} = -44.2177273205401$$
$$x_{28} = -38.2510541282376$$
$$x_{29} = 26.6498514574821$$
$$x_{30} = 16.9046192283974$$
$$x_{31} = -18.5109382379574$$
$$x_{32} = 92.3613326305303$$
$$x_{33} = -88.1102546346649$$
$$x_{34} = 66.4056178257571$$
$$x_{35} = -64.1508948593432$$
$$x_{36} = -12.7381124092375$$
$$x_{37} = -46.2084977857701$$
$$x_{38} = 32.5738300131898$$
$$x_{39} = -74.1308075062542$$
$$x_{40} = 88.3664310248785$$
$$x_{41} = 82.375018132477$$
$$x_{42} = 62.4157579453486$$
$$x_{43} = -86.1127864643158$$
$$x_{44} = -5.74301612101296$$
$$x_{45} = -28.3368875035311$$
$$x_{46} = -94.1032983570039$$
$$x_{47} = 18.8309297171705$$
$$x_{48} = -7.3115816730434$$
$$x_{49} = -36.2645449807319$$
$$x_{50} = -52.1849682625092$$
$$x_{51} = -90.1078338924361$$
$$x_{52} = -70.1381651356283$$
$$x_{53} = -80.1211308116339$$
$$x_{54} = 56.4337104086693$$
$$x_{55} = -82.1182155582792$$
$$x_{56} = 64.4105280865885$$
$$x_{57} = -16.5695229622524$$
$$x_{58} = -54.1782609089954$$
$$x_{59} = -20.4631983204608$$
$$x_{60} = 7.96154504225972$$
$$x_{61} = -68.1421630415863$$
$$x_{62} = 24.6837303756218$$
$$x_{63} = 76.3849716980399$$
$$x_{64} = 42.495785122687$$
$$x_{65} = 52.4480034075534$$
$$x_{66} = 40.5082410274504$$
$$x_{67} = 70.3966457620089$$
$$x_{68} = -96.1011702662716$$
$$x_{69} = -30.3153447286813$$
$$x_{70} = -100.004956913936$$
$$x_{71} = -10.8654506014968$$
$$x_{72} = 9.55624995284133$$
$$x_{73} = -9.04433982996113$$
$$x_{74} = 30.5957583046983$$
$$x_{75} = -60.1607682345028$$
$$x_{76} = -62.1556753178058$$
$$x_{77} = 38.5220226949901$$
$$x_{78} = 68.400998762587$$
$$x_{79} = -42.2278099160914$$
$$x_{80} = -56.1720223810675$$
$$x_{81} = 11.2997190058202$$
$$x_{82} = -32.296381804162$$
$$x_{83} = -98.005050700024$$
$$x_{84} = 98.3544708350782$$
$$x_{85} = 90.3638247874991$$
$$x_{86} = -66.1463989860564$$
$$x_{87} = 50.4560161379821$$
$$x_{88} = 80.3781688739502$$
$$x_{89} = -22.4235611844925$$
$$x_{90} = 36.5373529142981$$
$$x_{91} = -48.2000176225044$$
$$x_{92} = -84.1154372504332$$
$$x_{93} = 74.3886510740335$$
$$x_{94} = 44.4844728781857$$
$$x_{95} = -34.2795629354021$$
$$x_{96} = 60.4213396148336$$
$$x_{97} = -50.1921992657937$$
$$x_{98} = 46.4741540144396$$
Signos de extremos en los puntos:
(86.36915933121747, 2.19999350339782e-3165)
(14.999280926813473, 2.09565559302052e-85)
(78.38148241489469, 8.41491857024383e-2601)
(28.62085386236018, 1.27646902925939e-331)
(96.35666219146327, 2.76076533517936e-3949)
(58.42730978146217, 1.50071915735733e-1432)
(34.55450714941399, 2.88394870733637e-489)
(-92.10551712448779, 4.93107049926044e-3765)
(13.125084358924044, 3.8488532261596e-64)
(-14.643079787754933, 1.44494984454117e-106)
(-72.13438576692971, 3.54943884469736e-2323)
(48.4647034274775, 1.03060858789133e-978)
(20.771997137725517, 4.51585706397205e-170)
(54.44058971960544, 1.36190674118343e-1240)
(94.358947085372, 1.47931048865774e-3785)
(-40.23886926695423, 7.14237465404434e-739)
(-26.361571955102548, 1.98285032736697e-325)
(6.6601553469860315, 3.31624377683807e-14)
(22.72382448814687, 3.02041328683687e-205)
(72.39253649839898, 7.60745139429253e-2214)
(-58.166205240799194, 1.3357577777206e-1520)
(-78.12419338537786, 2.97691927111302e-2719)
(-76.1274147285399, 9.41004361378032e-2584)
(84.37201848717905, 5.00746161235727e-3019)
(-24.390134816046736, 2.90049108207911e-280)
(100.25503698772079, 9.01820658402517e-4279)
(-44.217727320540064, 2.86537302960703e-888)
(-38.25105412823759, 2.19060340917184e-669)
(26.64985145748212, 5.07566121646344e-286)
(16.904619228397404, 3.77140431345026e-110)
(-18.51093823795736, 1.28380640455433e-165)
(92.36133263053033, 2.65876592457153e-3625)
(-88.1102546346649, 7.23192087191618e-3449)
(66.4056178257571, 3.69972914878973e-1858)
(-64.15089485934324, 1.02554626780828e-1843)
(-12.738112409237461, 2.93364317966072e-82)
(-46.20849778577015, 3.52579812533857e-968)
(32.573830013189806, 3.04226883578918e-433)
(-74.13080750625416, 9.97828747176261e-2452)
(88.36643102487854, 3.24239822987928e-3315)
(82.37501813247704, 3.82344324594221e-2876)
(62.415757945348595, 2.0939627483844e-1638)
(-86.11278646431583, 5.38114367666287e-3296)
(-5.743016121012964, 4.87166345986923e-20)
(-28.33688750353106, 4.54422131417275e-374)
(-94.10329835700394, 2.50149086293248e-3928)
(18.830929717170516, 2.25847149427632e-138)
(-7.311581673043402, 2.70516492847735e-30)
(-36.26454498073191, 2.25353130419225e-603)
(-52.18496826250919, 9.35798702166659e-1229)
(-90.10783389243608, 3.26044494353328e-3605)
(-70.13816513562826, 4.23547523132339e-2198)
(-80.12113081163389, 3.15924039939545e-2858)
(56.433710408669306, 7.80566283555038e-1335)
(-82.11821555827918, 1.12470568666248e-3000)
(64.41052808658847, 1.5196805489091e-1746)
(-16.569522962252368, 2.35906496696053e-134)
(-54.17826090899544, 1.45792763158548e-1322)
(-20.463198320460812, 2.33529087565485e-200)
(7.9615450422597185, 2.43799889413567e-21)
(-68.14216304158627, 1.69544117354065e-2076)
(24.683730375621803, 6.76443124823753e-244)
(76.3849716980399, 2.42553373002255e-2468)
(42.495785122686975, 4.1932102554903e-748)
(52.448003407553365, 7.97093913434912e-1150)
(40.50824102745043, 3.49587592584824e-678)
(70.39664576200886, 8.27775276032343e-2092)
(-96.10117026627161, 4.25752808934263e-4095)
(-30.315344728681275, 3.49185858154282e-426)
(-100.00495691393613, 5.77319625326306e-4431)
(-10.865450601496772, 1.95184034871264e-61)
(9.556249952841327, 4.36365374055561e-32)
(-9.04433982996113, 4.15778080665637e-44)
(30.595758304698325, 1.07618858108817e-380)
(-60.16076823450285, 7.85543053683815e-1625)
(-62.15567531780577, 1.54969202491514e-1732)
(38.522022694990106, 9.7756029712129e-612)
(68.40099876258702, 3.0215026775798e-1973)
(-42.22780991609136, 7.81109664461969e-812)
(-56.17202238106753, 7.61933358820865e-1420)
(11.299719005820185, 2.30412888023818e-46)
(-32.29638180416196, 8.99774125888357e-482)
(-98.005050700024, 3.01824869641447e-4257)
(98.35447083507825, 1.72849516886926e-4116)
(90.36382478749914, 1.60307210276243e-3468)
(-66.14639898605638, 2.27667327527876e-1958)
(50.45601613798206, 1.56492284011286e-1062)
(80.37816887395016, 9.79337114815848e-2737)
(-22.42356118449253, 1.42195625644996e-238)
(36.5373529142981, 9.16841017071105e-549)
(-48.200017622504404, 1.45528052393883e-1051)
(-84.11543725043325, 1.343185762218e-3146)
(74.38865107403352, 2.34532983556296e-2339)
(44.48447287818566, 1.6870473225035e-821)
(-34.27956293540211, 7.77547381294577e-541)
(60.42133961483356, 9.67873326312275e-1534)
(-50.19219926579367, 2.0149011930674e-1138)
(46.4741540144396, 2.27670717977593e-898)
Intervalos de crecimiento y decrecimiento de la función:Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
La función no tiene puntos mínimos
La función no tiene puntos máximos
Decrece en todo el eje numérico