Can anyone help me prove the convexity of this rational function? The man who proved the convexity o

Oakey1w

Oakey1w

Answered question

2022-06-06

Can anyone help me prove the convexity of this rational function? The man who proved the convexity of function used these facts. But I don't know this fact is correct or not. Here are the facts and function:
1. As N increases, f(N) goes to infinity, That implies that there must be a minima (either at N=0 or somewhere else with a finite N)
2. There cannot be more than one (positive) minima since we're dealing with second order equation.
f ( N ) = + c N 4 + d N 3 + e N 2 + f N + g / + a N 2 + b N
a,b,c,d,e,f,g is constants. and N 1. I guess the second order equation means that between the leading coefficient of the numerator and the denominator is 2.
Are these facts correct? I think fact 1 is no problem, but fact 2 is correct or not.
I am waiting for any answers.

Answer & Explanation

Belen Bentley

Belen Bentley

Beginner2022-06-07Added 28 answers

Fact 2 seems incorrect (but I may not be understanding your notation). Local minima correspond to points where the derivative vanishes. For a rational function of the form p / q where p is of degree n and q is of degree m, the numerator of the derivative is a polynomial of degree n + m 1 and as such may have up to n+m−1 roots, which lead to at most n + m 1 possible local minima. In this case, there may be as many as 5 local minima.

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