[微積] 高微2題

[a88241050 (再回頭已是百殘身)]

1. Suppose that f:[a,b]→R is continuous on [a,b], if a<c<b, f'(x) exists

   for each x屬於(a,c)∪(c,b) and if lim  f'(x) exists in R, then f is
                                     x→c
   differentiable at c    (true or false?)




                    C_1     C_2            C_n
2.Assume that C_0+ ----- + ----- +......+ ----- = 0 ,
                     2       3             n+1

where C_0,C_1,C_2,...,C_n are real constants.


Show that the equation

    C_0+C_1*x+C_2*x^2+.....+C_n*x^n=0


has at least one real root in[0,1]



2題拜託各位了>///<

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※ 發信站: 批踢踢實業坊(ptt.cc) 
◆ From: 140.135.32.38
推 math1209    :1. 與 MVT 有關. 2. 與積分有關.
→ a88241050   :第一題稍微想通了,第二題還是不知道怎做..
→ mathblue    :2. [0,1]積分 去討論f正負
→ keroro321   :第2題令F(x)=C_0*x+(C_1*x^2)/2.+(C_n*x^n+1)/(n+1)
→ keroro321   :F(0)=F(1)=0
→ keroro321   :F'(a)=0 for some a in (0,1)
→ a88241050   :感謝keroro,我會了~~Orz感恩