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Question 1020046: Consider the function
f(x)=x^5-x^3
Determine the behaviors as x→∞ and as x→-∞
Find all zeros. What is the multiplicity of roots (zeros)?
When will the function cross the x-axis? When will it cross the y-axis?
Please help with this.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
\ =\ x^5\ -\ x^3) .
When  is positive, both  and  are positive. When is negative, both and are negative. But as gets bigger, gets bigger faster than , so for positive , ) increases without bound as increases without bound. And for negative , decreases without bound as decreases without bound.
\ =\ x^3(x^2\ -\ 1))
Hence if \ =\ 0) , then
 multiplicity 3
Or
(x\ -\ 1)\ =\ 0)
So
or
A function crosses the axis whenever . In your case the function crosses the axis three times, at -1, 0, and 1.
All functions defined at  cross the  axis when . If a function is not defined at , then the function does not cross the axis.
John
My calculator said it, I believe it, that settles it
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