# The convex function proof.

Can anyone help me prove convexity conditions? I know convexity conditions.

1. The second derivative function is greater 0
2. first order convexity conditions.
3. convex function conditions

Because my function is very complicated and have many variables, I can't use these conditions. But I found some facts by using numerical example. I need to prove convex function only between some ranges. I already know this range value. Between this ranges, the left-hand limit value is +∞ and the right-hand limit value is also +∞ and there is no zeros. By using only these facts, Does it satisfy the convex function? If not, What else conditions are needed? My goal is that finding minimal point in this range. I am waiting for any answers. It can be big help. Thank you.

The objective function is here

$$f(N)= \frac{E[PM]^2E[PM^2]N^4+2kE[PM]E[PM^2](+2E[PM]-W)N^3+(-2k^3𝜇W^2E[PM]+k^2W^2E[PM^2]-6k^2WE[PM]E[PM^2]+6k^2E[PM]^2E[PM^2])N^2+(2k^4𝜇W^3-k^4𝜇^2W^3𝜆+C^2k^4𝜇^2W^3𝜆-2k^4𝜇W^2E[PM]+2k^3W^2E[PM^2]+4k^3E[PM]^2-6k^3WE[PM]E[PM^2]+4k^3E[PM]^2E[PM^2])N+k^4W^2E[PM^2]-2k^4WE[PM]E[PM^2]+k^4E[PM^2]E[PM^2]}{N(N-\frac{k(W(1-𝜆/𝜇)-E[PM])}{E[PM]})(N-\frac{k(W(1-𝜆/𝜇)}{E[PM]})}$$ The N is equal or bigger to 1. Except N, All variables are bigger to 0. The interval is that $$0< N < \frac{k(W(1-𝜆/𝜇)-E[PM])}{E[PM]}$$

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You should clarify the definition of $f$. A little mathjax would go a long way... –  copper.hat Aug 29 '13 at 2:33
when you say a,b,c are variables, do you mean variables in N? Also, when you say you know that both the right and left hand limits go to infinity in some interval, what is that interval? –  Betty Mock Aug 29 '13 at 2:55
I revised my question. Maybe it can be helpful. Thank you. –  김헌길 Aug 29 '13 at 5:32