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Thursday, October 29, 2015

UTM Ideals Varieties and Algorithm - Chapter 1 Section 2 Exercise 2

Problem:


Solution:

Note that y2 is non-negative, therefore we only need to focus on range of x such that x(x1)(x2) is non-negative.

The only two such ranges is [0,1] and [2,+).

Next, we look at the range [0,1], the curve start from 0 and go back to 0, therefore it must has a turning point, taking the first derivative we get

f(x)=x(x1)(x2)f(x)=(x1)(x2)+x(x1)+x(x2)=(x23x+2)+(x2x)+(x22x)=3x26x+2=3(x22x+1)1

Therefore the curve turn at x=13+1=0.42....

Last but not least, the curve is growing cubic fast in [2,+), it is easy to sketch there.




It looks like a fish, isn't it?

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