Sanjo, ja mislim da si Ti jedini koji moze odgovoriti na ovo pitanje.

Slazem se s prof. Mirkom Diksicem.

Find the first derivative  (f'(x)=0) and  solve this equation to find the extreme point (x*)

, find the second derivatives (f''(x*)) if f''(x*)>0 , x* is minimum point and find f(x*)

if f''(x*)

Hello Abd, 

Your "answer' " assumes several things (knowing numerical values of  first and second derivatives) that results in elementary Calculus I answer. My question does not assume any of these and therefore it does not appear to be that trivial. Can you omit these

assumptions ? Regards, Sanjo

Let [a,b] denote the compact interval and let f be C1 on [a,b]. Let a

One can describe behavior of a C2 function f  around its root x* in the interior of a compact interval I using the integral F(x) from x* to an arbitrary x in I. We know that the derivative of F at the root x* is zero by the fundamental theorem of calculus. Now use the quadratic envelope characterization of zero derivative points (see earlier Q&A)  to show the following: F(x) is over-estimated on I  by the parabola c.(x-x*)^2, with the apex at x*, where c is half of the largest absolute value of the derivative of f on I. This behavior also holds around roots in the interior of I for C1 functions with Lipschitz derivative on I  . Thanks. Sanjo

I believe that your conditions: f in C2 or f in C1 with f' Lipschitz can be relaxed. The following refinement of my previous argument only assumes that f is C1 on [a,b].

Let L denote the infinity norm of f' on [a,b]. Let x0 in [a,b] be a zero of f. Let t be any number between x0 and x, then by the mean value theorem

abs(f(t))

Upon reflection I realized that it f in C1 is also too strict. It is enough that f is uniformly Lipschitz continuous on [a,b]. In this case we have a non negative real L such that

abs(f(x)-f(y))

Using the fundamental theorem of calculus and the quadratic envelope characterization of zero-derivative points, the following (visually pleasing) property of general C2 function f was proved around its root x* in the interior of a compact interval I in the domain of f: Integral from x* to an arbitrary point x in I is over-estimated by a parabola c.(x - x*)^2 on I, for a sufficiently large constant c, where x* is the apex of the parabola. Let us call this property "the quadratic envelope property" (abbreviated: QEP).

   Using the uniform bound characterization of zero-derivative points (see related Q&A), QEP can be stated differently: Absolute value of  Integral from x* to an arbitrary x in I ,different from x*, when divided by (x - x*)^2, is uniformly  bounded by a constant c around x*. Let us call this "the uniform bound property" (abbreviated: UBP).

Since quadratic envelope characterization of zero-derivative points extends to C1 functions with Lipschitz derivative, QEP also holds for such functions. Carl Christian has shown in his answers above that the Lipschitz derivative assumption can be released in QEP. This would mean that QEP holds for every C1 function. In view of his comments GEP also holds for uniformly Lipschitz continuous functions.

His latter comment raises the question of identifying classes of meaningful functions for which QEP holds. As f(x) = 2/3 times x^(3/2) shows, a major problem here could  be determination of the constant c. Thanks, Sanjo

.

Of course, when QEP and UBP are applied to the derivative of f, these properties recover necessary conditions for optimal points of f.

Carl. Christian noted above that QEP holds around roots x* of C1 functions. A different proof  follows from the observation that F'' = f' and the quadratic envelope characterization of zero derivative point applied to F'(x*) = 0. (Here  F(x) is the integral of f from  x* to x.)

Here is a complete formulation of the quadratic envelope property (QEP). For a function of the single variable x, QEP is an inequality in two variables x and x*. It recovers some of the old and introduces new properties of f.

Theorem Consider a C1 function f of the single variable x  and a point x* in the interior  of a compact interval I in its domain. For every x in I the following holds

(QEP)    abs [F(x,x*) - f(x*)(x - x*)] is overestimated by c (x - x*)^2.

Here F(x,x*) = integral of f from x* to x and constant  c = half of max [abs f"(t) : t in I].

 Proof: It follows from the fundamental theorem of calculus and the characterization of  zero-derivative point.

Let us call a relation between x and x*, that follows from (QEP), "augmentor" of f on I.

Illustrations: (a) Trivial augmentor. For f(x) = x  and any compact interval I, (QEP) yields, after multiplication by 2,  abs (x^2 - x*^2 - 2x* (x - x*)

Following the above, it appears that for C1 functions on compact intervals with interior points one can estimate the  areas under curves and the average values of functions by differentiation instead of   integration ! Have these approaches been  generally  used  when finding integrals is hard?

Hi, as shown in the linked publication, one can obtain approximations for roots not only from applications of the Newton theorem for the function f, but also by application of the Newton theorem for its inverse function. Using the integral calculus mean value theorem, one can also obtain expressions for x* that are in principle exact.  For instance, assuming f(x*)=0, one has

f(x)=int(f'(s),s=x*..x)

     = (x-x*) int(f'(t x + (1-t) x*),t=0..1)

     = (x-x*) f'(x* + tau(x,x*) (x-x*))

with 0< tau(x,x*) < 1 by the mean value theorem for sufficiently smooth f. This is one of the "in principle exact" expressions. A rather good approximation for x* results using the midpoint rule, i.e. tau(x,x*) = 1/2. I guess that combining the above with your approach may allow to control the error of this and related approximations.  Since you noted difficulties for f ~ (x-x*)^{3/2} that does have vanishing right derivative at x* and hence, no inverse around x*, it might be helpful to extend your investigation also to related properties of the inverse functions and the corresponding "in principle exact" formula. Probably, this will allow to find classes of functions where a combined approach is working well.

Article On Newton-type methods with cubic convergence

Hello Herbert,

Thank you very much  for your original comments. A  straightforward application of my research is (QEP), given in the above Theorem. It gives a bound on integrals by derivatives and vice versa. (Could these be some of the Ostrowski inequalities ?)

Note that the constant c is introduced as the maximal absolute value of f"(t) on I. This may be related to your "tau" and control of the error of approximation !?  It would be great if (QEP) could somehow be extended to your results and  related properties of inverse functions. By all means, these appear to be new  promising directions !

The above answers have resulted in the joint  technical report:"Bounding integrals by derivatives: illustrations in the study of tumor budding"  available on RG.

Using FTC and quadratic envelope characterization of zero-derivative points it is easy to show that the roots f(x*) = 0   of continuous (not necessarily differentiable) Lipschitz function f  in the interior I of domain of single variable functions are apexes of a class of parabolas over-estimating the absolute value of integral of f from x* to any x on I. A related result using apexes  characterizes fixed points f(x*) = x* and it is recently  proved in the author's research report in Nov. 2017. Interestingly, there are also dual versions of this result.

Theorem:(Primal Necessary Condition for a Root f(x*) = 0.) Consider a continuous Lipschitz function f of the single variable x on  I =  [a, b] and a point x* in the interior of I..  Denote V(x) = integral of f(t) on [x* , x] from x* to x in I..

If f(x*) = 0 than   abs V(x) is less than or equal to c times (x - x*)^2,  x in I, for some non-negative  constant  c.

(Remark:The attached file is a technical report from recent Researchgate..)

Overall conclusion: Point x* such that derivative  f'(x*) = 0, root f(x*) = 0,or  fixed point     f(x*) = x*,  is an apex of a class of parabolas. The first statement holds for C1 functions in n variables with Lipschitz derivative, the latter two are proved for continuous single variable (not necessarily differentiable) Lipschitz functions using FTC and/or quadratic envelope property. Each of the three also has a  "dual" formulation.

Two characterizations ("primal" and "dual") of roots of a continuous Lipschitz function  f are given in the file "Essence of roots", which is an addendum to the current  RG project "Studying the essence of fixed points". The characterizations require only continuity and Lipschitz property of function f (no differentiability !) The fundamental theorem of calculus is used thereby when working with a particular integral.. This may be an answer to  the posed question. 

There are straight forward answers to characterizing f(x*) = 0 given in the current RG project "Studying essence..." See attached file.

There are straight forward answers to characterizing f(x*) = 0 given in the current RG project "Studying essence..." See attached file.

The above shows that the points called "roots" x* can be characterized using properties of f(x) around x*, not only f(x*) = 0.

The above shows that the points called "roots" x* can be characterized using properties of f(x) around x*, not only f(x*) = 0.

The above results show that the roots x* depend on the bahavior of f around x* not only on f(x*) = 0.

The above results show that the roots x* depend on the bahavior of f around x* not only on f(x*) = 0.

The roots depend on the bahior of f around them.

There are at least two different formulations of roots.

There exist different formulations of roots.