@ANANDMUNAGALA   Jacobi's method requires only the simplest of the rotation matriсes, such as those

See the solution http://www.mapleprimes.com/questions/200181-What-Is-The--Maple-Procedure-To-Find

@brian bovril  We must prove that outside a bounded interval there are no roots. It is not enough to refer to the plot of the function.

@Carl Love  Thanks a lot for the solution of the problem!

@Carl Love In Maple 12 (Classic) and Maple 16 (Standard) the result is correct. May be this is a bug in Maple 17?

1) The  simplex[convexhull]  command can be adapted to the verification of the polygon on the convexity. But it still solves another problem and works only with data type numeric. It is not always convenient. See the screenshot.

2) The  IsSimple  procedure solves your example correctly. See the screenshot.

see  http://www.mapleprimes.com/posts/200064-Checking-Of-A-Polygon-On-Convexity-

@Carl Love  Of course, you're right! The range  [-11, -9]  is not included in the domain of the function. We have

 Range( limit(sqrt((x^2-5*x+6)/log[10]((x+10)^2)), x = 2),  limit(sqrt((x^2-5*x+6)/log[10]((x+10)^2)), x = -9, right)); 

                                                                 Range(0,  infinity)

I like Axel's code more than mine, because it automatically divides the sum by two parts depending on  x  and is suitable for various  x. I only suggest to simplify it:

g:= x ->
    Sum(sin(x/(k+1))/k, k = 1 .. floor(x)) + Sum(sin(x/(k+1))/k, k = 1+floor(x) .. infinity);
    
plot(g, 0 .. 100);

depends(sin(x)^2+cos(x)^2,  x);

                           true

@Carl Love  Thanks for the useful remark!

@Preben Alsholm 
Thank you for your useful suggestion to improve the code!

What is to be printed, if the 4th point satisfies exactly 2 of these inequalities?

@Carl Love 

Thanks for the link! I did not know about this example in the Maple help. In my post slightly more general problem is solved:  m  queens on an  n×n  chessboard.

1. The branch-and-bound is applied. Tree of variants is constructed , starting with the first vertical. Subprocedure  It  prolongs each branch further to 1 step, cutting branches worthless.

2.   The goal of the post - to show the possibilities of Maple for programming such problems.

@Markiyan Hirnyk 
Thanks for the feedback! All of these methods seem to me too complicated and not of fundamental importance for the problem.