Учебно-методическое пособие (1017796), страница 9
Текст из файла (страница 9)
S ′ (a) > 0S ′ (a) = 0 ïðè a1,2 = p2√√2− 22−2ïðè a < pè S ′ (a) < 0 ïðè a > p, ñëåäîâàòåëüíî,22103ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ√2− 2a1 = p òî÷êà ëîêàëüíîãî ìàêñèìóìà.  òî÷êå a3 = p2 ′ïðîèçâîäíàÿ S (a) íå ìåíÿåò çíàê, ñëåäîâàòåëüíî, a3 = p íå ÿâëÿ√2+2åòñÿ òî÷êîé ëîêàëüíîãî ýêñòðåìóìà.
S ′ (a) < 0 ïðè a < p√√22+22+2è S ′ (a) > 0 ïðè a > p, ñëåäîâàòåëüíî, a2 = p22òî÷êà ëîêàëüíîãîìèíèìóìà. Íàéäåì√√ äëèíó âòîðîãî êàòåòà b ïðèp(2a − p)2− 22− 2a=p:b==p. Ñëåäîâàòåëüíî, èñêîìûé22(a − p)2òðåóãîëüíèê ðàâíîáåäðåííûé. Åãî ïëîùàäü(√ )2)22ab p2− 2p2 ( √S=2−1==2224Îòâåò: èç âñåõ ïðÿìîóãîëüíûõ òðåóãîëüíèêîâ ñ çàäàííûì ïåðèìåòðîì íàèáîëüøóþ ïëîùàäü èìååò√ ðàâíîáåäðåííûé òðåóãîëüíèê ñ äëèíîé êàòåòîâ a = b = p)2p2 ( √2−1 .S=42− 2. Èñêîìàÿ ïëîùàäü ðàâíà26. Èññëåäîâàíèå ôóíêöèè: âûïóêëîñòü è âîãíóòîñòü,àñèìïòîòû6.1. Âûïóêëîñòü è âîãíóòîñòü ãðàôèêà ôóíêöèèÎïðåäåëåíèå 6.1 Ôóíêöèÿ f (x) íàçûâàåòñÿâûïóêëîé âíèç(èëè ïðîñòî âûïóêëîé) íà èíòåðâàëå (a, b), åñëè ãðàôèê ôóíêöèèy = f (x) èä¼ò íå âûøå õîðäû, ñîåäèíÿþùåé ëþáûå äâå òî÷êè ãðàôèêà (x0 , f (x0 ) è (x1 , f (x1 ) ïðè x0 , x1 ∈ (a, b).Àíàëîãè÷íî, ôóíêöèÿ f (x) íàçûâàåòñÿ âûïóêëîé ââåðõ (èëè âîãíóòîé) íà èíòåðâàëå (a, b), åñëè ãðàôèê ôóíêöèè èä¼ò íå íèæåõîðäû, ñîåäèíÿþùåé ëþáûå äâå òî÷êè ãðàôèêà.104y6y6................................................................................................................................................................................................................y=f(x)y=f(x)--xÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀxÐèñ.
14. Âûïóêëàÿ ôóíêöèÿÐèñ. 15. Âîãíóòàÿ ôóíêöèÿÒåîðåìà 6.1 Ïóñòü íà èíòåðâàëå (a, b) ôóíêöèÿ y = f (x) èìå-åò âòîðóþ ïðîèçâîäíóþ f ′′ (x). Ôóíêöèÿâûïóêëàíà (a, b) òîãäàè òîëüêî òîãäà, êîãäà f ′′ (x) > 0 ïðè âñåõ x ∈ (a, b), èâîãíóòàòîãäà è òîëüêî òîãäà, êîãäà f ′′ (x) 6 0 ïðè âñåõ x ∈ (a, b).y6............................................................................................................................................................................................................................′′...... ′′.......00..............′′..............0.........................
........... ....... ........ ............... ... ............... ...............................................................................................y=f(x)f (x) >f (x) 60Ðèñ. 16.f (x) >x1056.2. Òî÷êè ïåðåãèáà.ôóíêöèè f (x) íàçûâàåòñÿòàêàÿ òî÷êà x0 ∈ (a, b), â êîòîðîé âûïóêëîñòü ñìåíÿåòñÿ íà âîãíóòîñòü. Äðóãèìè ñëîâàìè, òî÷êà ïåðåãèáà x0 ∈ (a, b) ðàçäåëÿåò íåêîòîðóþ δ -îêðåñòíîñòü òî÷êè íà äâà èíòåðâàëà (x0 −δ, x0 )è (x0 , x0 +δ), íà îäíîì èç êîòîðûõ ôóíêöèÿ âûïóêëà, à íà äðóãîì- âîãíóòà.Òî÷êîé ïåðåãèáàÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀÎïðåäåëåíèå 6.2y6............................................................................................. .................... ...............................................................................y=f(x)âûïóêëàâîãíóòà0x0-xÐèñ.
17.Íåîáõîäèìîå óñëîâèå òî÷êè ïåðåãèáàÒåîðåìà 6.2 Ïóñòü x0 ∈ (a, b) òî÷êà ïåðåãèáà ôóíêöèè f (x)è ñóùåñòâóåò f ′′ (x0 ). Òîãäà f ′′ (x0 ) = 0.Òàêèì îáðàçîì, åñëè x0 ∈ (a, b) òî÷êà ïåðåãèáà, òî ëèáîf ′′ (x0 ) = 0, ëèáî f ′′ (x0 ) íå ñóùåñòâóåò (â ÷àñòíîñòè, f ′′ (x0 ) = ∞).Ïðèâåäåì ïðèìåðû.106y6............................. ′′............................................................................................................................................ ′′.....0.................y=f(x)f (x) > 0xÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ0f (x) <Ðèñ. 18.Òî÷êà 0 òî÷êà ïåðåãèáà ôóíêöèè f (x) = x3 .
Çäåñü f ′′ (0) = 0.y6y=f(x).....................................................................................................................................′′......0.......................................................′′.............0.............................................................................................................................f (x) >0xf (x) <Ðèñ. 19.Òî÷êà 0 òî÷êà ïåðåãèáà ôóíêöèè f (x) = x3 . Çäåñü f ′′ (0) = ∞.107y6..............................................
′′................................................................................................-2 ...............................................y=f(x)f (x) = 2xÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀ0f ′′(x) =Ðèñ. 20.Òî÷êà 0 òî÷êà ïåðåãèáà ôóíêöèè f (x) = x2 sign x. Çäåñüf ′ (x) = 2|x|, è, ñëåäîâàòåëüíî, f ′′ (0) íå ñóùåñòâóåò. Íàïîìíèì,÷òî−1, x < 0sign x =0, x = 01, x > 0Äîñòàòî÷íîå óñëîâèå òî÷êè ïåðåãèáàÒåîðåìà 6.3 Ïóñòü f (x) èìååò âòîðóþ ïðîèçâîäíóþ f ′′ (x) âíåêîòîðîé îêðåñòíîñòè òî÷êè x0 , íåïðåðûâíóþ â ýòîé òî÷êå.Åñëè f ′′ (x0 ) = 0 è ïðè ïåðåõîäå ÷åðåç òî÷êó x0 âòîðàÿ ïðîèçâîäíàÿ f ′′ (x) ìåíÿåò çíàê, òî òî÷êà (x0 , f (x0 )) ÿâëÿåòñÿ òî÷êîéïåðåãèáà ãðàôèêà ôóíêöèè f (x).Ïðèìåð 6.1. Íàéòè èíòåðâàëû âûïóêëîñòè è âîãíóòîñòè ôóíêöèè f (x) = x4 − 2x2 . Óêàçàòü òî÷êè ïåðåãèáà.108Ðåøåíèå: Âòîðàÿ ïðîèçâîäíàÿ f ′′ (x) = 12x2 − 4. f ′′ (x) ðàâíàíóëþ â òî÷êàõ√√33è x1 =.33×òîáû íàéòè èíòåðâàëû âûïóêëîñòè, ðåøèì íåðàâåíñòâîf ′′ (x) > 0.
Ðåøåíèåì ÿâëÿåòñÿ îáúåäèíåíèå èíòåðâàëîâ()√ ) (√33x ∈ −∞; −∪; +∞ .33ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀx0 = −Äëÿ íàõîæäåíèÿ èíòåðâàëà âîãíóòîñòè íóæíî ðåøèòü íåðàâåíñòâî f ′′ (x) < 0. Ðåøåíèåì ÿâëÿåòñÿ( √ √ )3 3.x∈ − ;3 3Ïîëó÷åííûå äàííûå çàíîñèì â òàáëèöó. Íà îñíîâàíèè èçìåíåíèÿçíàêà âòîðîé ïðîèçâîäíîé äåëàåì âûâîä, ÷òî òî÷êè√3x0 = −3√3è x1 =.3ÿâëÿþòñÿ òî÷êàìè ïåðåãèáà.(√ )3x ∈ −∞; −3√3x=−( √ 3√ )3 3x∈ − ;3 3√3x=3f ′′ (x)f (x)+âûïóêëà (âíèç)0òî÷êà ïåðåãèáà−âîãíóòà (âûïóêëà ââåðõ)0òî÷êà ïåðåãèáà109(√)3x∈; +∞3f ′′ (x)f (x)+âûïóêëà (âíèç)Ïîâåäåíèå ôóíêöèè ìîæíî ïðîèëëþñòðèðîâàòü ñëåäóþùèìãðàôèêîì.y=x4 − 2x26ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀy..............................................................................................................33.................1.1....3..3............................................................................................ 0........................................ .........
.......................................................................................................................................... ............................. . ........ ... ...................... ....... ....... ....... ....... .......1....... .......
....... ....... ....... .................................√√−−-x−Ðèñ. 21.6.3. Àñèìïòîòû ãðàôèêà ôóíêöèèÀñèìïòîòîé êðèâîé íàçûâàåòñÿ ïðÿìàÿ, ðàññòîÿíèå äî êîòîðîé îò òî÷êè, ëåæàùåé íà êðèâîé, ñòðåìèòñÿ ê íóëþ ïðè íåîãðàíè÷åííîì óäàëåíèè îò íà÷àëà êîîðäèíàò ýòîé òî÷êè ïî êðèâîé. Âçàâèñèìîñòè îò ïîâåäåíèÿ àðãóìåíòà ïðè ýòîì, ðàçëè÷àþòñÿ äâàâèäà àñèìïòîò: âåðòèêàëüíûå è íàêëîííûå. Äëÿ óäîáñòâà ñôîðìóëèðóåì îòäåëüíî îïðåäåëåíèÿ âåðòèêàëüíîé è íàêëîííîé àñèìïòîò.110ãðàôèêà ôóíêöèèy = f (x) íàçûâàåòñÿ ïðÿìàÿ x = a, åñëè f (x) → +∞ èëèf (x) → −∞ ïðè êàêîì-ëèáî èç óñëîâèé: x → a+, x → a−, x → a.Îïðåäåëåíèå 6.3Âåðòèêàëüíîé àñèìïòîòîéãðàôèêà ôóíêöèèy = f (x) ïðè x → +∞ íàçûâàåòñÿ ïðÿìàÿ y = kx + b, åñëèÎïðåäåëåíèå 6.4Íàêëîííîé àñèìïòîòîélim [f (x) − (kx + b)] = 0.ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀx→+∞Òàêèì îáðàçîì, ñóùåñòâîâàíèå íàêëîííîé àñèìïòîòû y = kx+bó êðèâîé y = f (x) ïðè x → +∞ îçíà÷àåò, ÷òî äàííàÿ ôóíêöèÿ ïðèx → +∞ âåäåò ñåáÿ ïî÷òè êàê ëèíåéíàÿ ôóíêöèÿ, ò.å.
îòëè÷àåòñÿîò ëèíåéíîé ôóíêöèè y = kx + b íà áåñêîíå÷íî ìàëóþ ïðè x →+∞.Àíàëîãè÷íî îïðåäåëÿåòñÿ íàêëîííàÿ àñèìïòîòà ïðè x → −∞. ñëó÷àå, åñëè k = 0, íàêëîííàÿ àñèìïòîòà íàçûâàåòñÿ ãîðèçîíòàëüíîé. Òàêèì îáðàçîì, ïðÿìàÿ y = b - ãîðèçîíòàëüíàÿ àñèìïòîòà ïðè x → +∞ (x → −∞), åñëè lim f (x) = b èëè lim f (x) = bx→+∞x→−∞ñîîòâåòñòâåííî.Òåîðåìà 6.4 Ïðÿìàÿ y = kx + b ÿâëÿåòñÿ íàêëîííîé àñèìïòî-òîé äëÿ ãðàôèêà y = f (x) ïðè x → +∞ òîãäà è òîëüêî òîãäà,êîãäàf (x)k = lim, b = lim [f (x) − kx]x→+∞ xx→+∞(ñîîòâåòñòâåííî ïðè x → −∞ , êîãäàf (x),x→−∞ xk = limb = lim [f (x) − kx] .x→−∞Äëÿ íàõîæäåíèÿ íàêëîííîé àñèìïòîòû íóæíî ñíà÷àëà íàéòèk , ò.å. âû÷èñëèòü ïåðâûé èç óêàçàííûõ ïðåäåëîâ. Åñëè ýòîò ïðåäåë íå ñóùåñòâóåò, òî íàêëîííîé àñèìïòîòû ó ãðàôèêà íåò.
Åñëè111ïðåäåë ñóùåñòâóåò (k < ∞) , òî çàòåì âû÷èñëÿåòñÿ b. Åñëè êàêîéëèáî èç ýòèõ äâóõ ïðåäåëîâ íå ñóùåñòâóåò, òî íåò è íàêëîííîéàñèìïòîòû.Ïðèìåð 6.2. Íàéòè àñèìïòîòû ãðàôèêà ôóíêöèè f (x) =1.x−1Ðåøåíèå: Ãðàôèê y = f (x) èìååò âåðòèêàëüíóþ àñèìïòîòó x = 1,ïîñêîëüêó lim f (x) = ∞, è ãîðèçîíòàëüíóþ àñèìïòîòó y = 0, ò.ê.x→1x→∞ÊàÌôåäÃÒðÓàÌ ÂÌÈ-2ÐÝÀlim f (x) = 0.y6....................................................................................................1..............
0..................................................................................................................................................1...............1...................................................................................................................................................y=(x − )-xÐèñ. 22.Ïðèìåð 6.3. Íàéòè àñèìïòîòû ãðàôèêà ôóíêöèèf (x) =x1+√ .2xÐåøåíèå: Ãðàôèê ýòîé ôóíêöèè èìååò âåðòèêàëüíóþ àñèìïòîòóxx = 0, ïîñêîëüêó lim f (x) = +∞, è íàêëîííóþ àñèìïòîòó y =2ïðè x → +∞ (âû÷èñëèòü êîýôôèöèåíòû k è b ñàìîñòîÿòåëüíî).x→0+112y...6.................................
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
