16.函数极值点与拐点的一种判别法.pdf

38 4 2009 M7 内蒙古师范大学学报(自然科学汉文版) Journalof Inner Mongolia Normal University(Natural Science Edition) Vol.38No.4 July 2009 l :2008-10-19 “: 81 S “(M103c83) Te: (1971-), o, 8 4 g , =,1V Y #m ,E-mail:lin rongfei163.com. f .B YE 林荣斐,林炳江 (台州学院数学与信息工程学院,浙江临海 317000) K 1:f .B YZE.BHq/, f(n)(x)x0 #U0-(x0) U0+(x0)|s, x0 wLy =f(x),(x0, f(x0) wLy=f(x).,i . 1oM:;.; f ; YE ms |:O172 DS M :A cI|:1001-8735(2009)04-0400-03 0 引言 11 !f y =f(x)x0 V,U0(x0 ,) = V, Of(x0)=0, U0-(x0) U0+(x0) =f(x)|M,5(x0,f(x0) wLy =f(x)B. 21 !f y =f(x)x0 #U0(x0 ,) = V, Of(x0)=0, U0-(x0) U0+(x0) =f (x)|M,5(x0,f(x0) wLy =f(x)B. 32 !f y =f(x)x =x0 # = n (n 2), Tf(x0)= f(x0)=f(n-1)(x0)=0,7f(n)(x0)0, * : n H,(x0,f(x0) wLy =f(x).; n H,x =x0 wLy =f(x) Of(n)(x0)0 H,x =x0 l, f(n)(x0)0. x U0-(x0),5f(n)()0, (x-x0)n-2 0,(x-x0)n-2 0, f(x)0.N V (x0 ,f(x0) wLy =f(x). (2)n H,|y =f(x)x0)Z 7n-1 T f(x)=f(x0)+f(x0)(x-x0)+f(x0)2! (x-x0)2 + f(n-1)(x0) (n-1)!(x-x0) n-1 +f(n)() n! (x-x0) n, xx0W.f(x0)=f(x0)=f(n-1)(x0)=0, f(x)-f(x0)=f (n)() n! (x-x0) n. yU0-(x0)U0+(x0) =f(n)(x)|M,“ !f(n)(x)0. xU0-(x0),5f(n)()0, (x-x0)n 0, f(x)-f(x0)0,f(x)f(x0). x U0+(x0),5f(n)()0,(x-x0)n 0, f(x)-f(x0)0,f(x)f(x0).N Vx =x0 wLy =f(x), Ol. ,f(n)(x)0 H,x =x0 wLy =f(x), Ov. V, 4/w . w 1 !f y =f(x)x0 #U0(x0) n(n 2) , Of(x0)= f(x0)=f(n-1)(x0)=0. U0-(x0)U0+(x0) =f(n)(x)|M,5: (1)n H,x =x0 wLy =f(x); (2)n H,(x0,f(x0) wLy =f(x). n H,x =x0 wLy =f(x),5U0-(x0)U0+(x0) =f(x)|MQ, f(x)U0(x0) =,5f(x)U0(x0) =|,(x0,f(x0) wLy=f(x). . V , n H,(x0 ,f(x0) wLy =f(x). 2 !f y =f(x)x0 #U0(x0) n , Of(x0)=f(x0)= f(n-1)(x0)=0. U0-(x0)U0+(x0) =f(n)(x)|MQ,5: (1)n H,x =x0 wLy =f(x); (2)n H,(x0,f(x0) wLy =f(x). V 1 . w 2 !f y =f(x)x0 #U0(x0) n , Of(x0)=f(x0)= f(n-1)(x0)=0. U0-(x0)U0+(x0) =f(n)(x)|MQ,5: (1)n H,(x0,f(x0) wLy =f(x).; (2)n H,x =x0 wLy =f(x). 2 极值点与拐点的不重合性 l3 , USx0x ,.(x0,f(x0) wL . ZL, wL ,9 wL . . YE V, V ? (9)B i(Al), V ?. = , #= i(Al). Bf B)B= ( ,、B ,7= iB= ( i f ( V ? 3, :y =x3,x=0y=0,y=0;y =x75 ,x=0y=0,yi ;y=x35x=0yy (i.yf(x0)=limxx 0 f(x)-f(x0) x-x0 , i“f ,x0 f(x0)i,f(x0)i. / 3 f x0 V ? H. 401 = =Sv(1 Sq)38 f1 f y =f(x)U(x0 ,) = = , Of(x0)=0,f(x0)=0.A ,f m ; , ML, f /x0 V ?, H9 . V ?. (1)x0f(x),N HU0-(x0),U0+(x0) f(x)|MQ,5w 2,(x0,f(x0) wLy =f(x). (2)(x0 ,f(x0) f(x).,N HxU(x0,),f(x)U0-(x0),U0+(x0) |MQ,w 2 ,x0 f y =f(x). f2 f y =f(x)U(x0 ,) =B V,U0(x0) = V, Of(x0)=0,f(x0)i, A ,f m ;, ML, f /x0 V ?, H9 . V ?. f3 f y =f(x)U0(x0) = V,f(x)f(x0) (i. ,N Hf(x) i, wLx0) x ML( ),5 ; wL,V7., 1 f .N Hx0 V ?, H9 . V ?. , f1,w 2 V . V ?. ID: 1 .1 wL. YEJ. Ll M,2003,33(1):99-101. 2 I .i sHqJ. l jv,1998,9(3):72-73. 3 ;.l#.J . S .2002.6,18(3):96-99. The Judging Methodsof Extreme Point and Inflection Point LIN Rong-fei, LIN Bing-jiang (Schoolof Mathematics and Information Engineering,TaizhouUniversity,Linhai 317000,Zhejiang,China) Abstract:A new wayispresented to judgewhether x0 is theextremepoint or an inflection point of a curve y=f(x).Under the certain condition,we use the sign of f(x)in U0-(x0)and U0+(x0)to judge whether acurvey=f(x)hasextremevalueon x0 or(x0 ,f(x0)is an inflection point of y=(f(x).Wealso explain thenon-coincidence of theextremepoint and theinflection point of y=(f(x). Key words:extremepoint;inflection point;continuous function;judging method 【责任编辑陈汉忠】 (上接第399页) An Identity Involving thek-power Complement Number Sequence GUO Yan-chun, ZHENG Ya-ni (Departmentof Mathematics,Xianyang TeachersCollege,Xianyang 712000,Shaanxi,China) Abstract:For any positiveintegerk2 and any positive integer n,we call ak(n)as the Smarandache k-th power complement number of n,if ak(n)is thesmallest positiveinteger such that nak(n)is a perfect k-th power number.That is,ak(n)=minu:un=mk;u,mN.The main purpose of this paper is to study theconvergent property of theseries + n=1 1 (nak(n)s using theanalyticmethod,andgivean interesting identity. Key words:k-th power complement number;infiniteseries;identity 【责任编辑陈汉忠】 402