Differentiation + integration

2010-01-24 10:27 pm

1. Find the range of values of p for which the curve f(x)=x^3-3x^2+2x has 3 different tangents passing through point (0, p).

2. Please refer to this link:
http://i1013.photobucket.com/albums/af256/wingkei123/19a0fe4f869ec4b82f44a57d7c4bf714.png

更新1:

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更新2:

請問Discriminant=18abcd-4a^3d+b^2c^2-4ac^3-27a^2d^2點得黎架???

回答 (1)

用戶9112

2010-01-25 7:22 am

✔ 最佳答案

Please see the solution below:

圖片參考：http://img31.imageshack.us/img31/4945/qq1e.png

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圖片參考：http://img215.imageshack.us/img215/6631/qq2.png

2010-01-25 00:06:19 補充：
http://img31.imageshack.us/img31/4945/qq1e.png
http://img215.imageshack.us/img215/6631/qq2.png

2010-01-25 21:23:44 補充：
(1)我做錯了最後一步,
p(1 – p) > 0
p(p – 1) < 0
0 < p < 1
(2) Discriminant 請參考http://en.wikipedia.org/wiki/Cubic_equation

2010-01-25 21:23:55 補充：
(3) 另有做法
Let g(a) = 2a^3 – 3a^2 + p = 0
g'(a) = 6a^2 – 6a
g”(a) = 12a – 6
g’(a) = 0 => 6a(a – 1) = 0 => a = 0 or a = 1
g”(0) = -6 => max point (0, p)
g”(1) = 6 => min point (1, p – 1)

2010-01-25 21:24:00 補充：
If max y > 0 and min y < 0, then the equation has 3 roots (you can draw the graph to convince yourself).
p > 0 and p – 1 < 0 => 0 < p < 1

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