Variation Question: f Min then \delta f, \delta^2 f?

[matematikuvol]

If f has minimum, than

$$\delta f=0$$, $$\delta^2 f>0$$

or

$$\delta f>0$$?

 

[Char. Limit]

I think you're going to need to answer your question more clearly. I really am not sure what you're talking about.

 

[matematikuvol]

If some function has minimum. Then variation of that function is equal zero or bigger then zero?

 

[HallsofIvy]

Well, what do you mean by "variation" of a function? I know the "total variation of a function" on a given interval. If that is what you mean, what is the interval.

(The derivative, if that might be what you mean, of a function, at a minimum, is 0 and the second derivative either 0 or positive. A simple example is [itex]x^2[/tex] which has a minimum at x= 0. The derivative is 2x which is 0 at x= 0 and the second derivative is 2 which is positive.)

 

[matematikuvol]

I mean by variation infinitesimal change of function while argument stay fiksed.

$$\varphi(x)$$
$$\bar{\varphi}(x)$$

variation

$$\delta \varphi(x)=\bar{\varphi}(x)-\varphi(x)$$

 

[HallsofIvy]

Which again makes no sense because you don't say how φ¯(x) is related to φ(x).