Maximization third degree function

mnienke

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Dec 28, 2013
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I have the following function I want to maximize over I:
(1+k)^2/4-k-0.5(1-k)kI/(I+eps)+0.25(kI/(I+eps))^3-I
With 0<k<=1, 0<eps, 0<I.

With the first order condition I get the following:
k*eps*I+k*eps^2-k^2*eps^2-2(I+eps)^3=0

I had hoped to be able to now write something like I=... as a function of k and eps, but this is not possible.
So I wanted to know for which values of k and eps you'll get that I=0 is maximal and for which values of k and eps I>0 is maximal.
But how?
 
Hmm, yeah I figured something like that out already but then without the substituting. Still thanks for the effort!
Still it is a very complicated equation and for a lot of values for 0<k=<1 and 0<eps, it gives complex values or does not really give me the right values.

I was more thinking of something like this:
With the second derivative I can get the inflection points: -eps +- sqrt(1/6k*eps).
There are indeed 3 solutions and for the maximum it has to hold that I>-eps+sqrt(1/6k*eps)

In this way I was trying to find some constraints for I, k and eps but I am sure that this is not all...
 
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