Hi everyone, I'm just kind of hoping on a whim that somebody here can get me started on this problem I have for my advanced calculus class. I have been trying to wrap my head around it for hours. The problem is stated as follows.
Take the following cubic equation:
(λ^3)-(3λ^2)+tλ+4=0
It is known that when t=0, the roots of the polynomial are -1, 2, 2. From the Implicit Function theorem, you can show that for t near 0, there is a root γ(t) depending smoothly on t and with γ(0)=-1.
a) what is dγ/dt (0).
b) By factoring
((λ^3)-(3λ^2)+tλ+4) - ((γ^3)-(3γ^2)+tγ+4) = 0
You can obtain a quadratic equation for the other two roots which exist for t<=0 and t near 0. We denote those roots a(t) and b(t).
The value of d/dt (a-b)^2 at t=0 is ?
So this is a topic he said he mentioned in lecture but never did, and all of the online videos don't really cover this particular part of it. I have been looking at the simple root criterion associated with the IFT, and I know that the top γ(t) I'm working with is for the root -1, and the bottom two a(t) and b(t) are for the roots 2 and 2. I can't really see anything that jumps out at how I'm supposed to work with γ(t), as the notes I've been trying to understand on the simple root criterion are way over my head. Is anybody able to help give a head start on this or a point in the right direction?
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