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These are replies submitted by Kitonum

@acer  This example shows the importance of visualization (a plot) in these methods. How else would we know a priori that the result

Optimization:-Maximize(f, x=0..2*Pi, method = branchandbound);

is incorrect?

If we want to solve purely programmatically (not looking at the plots), then probably the best method will be a standard approach using the first derivative:

R:=[RootFinding:-Analytic(diff(f(x),x), x, re=0..2*Pi, im=-1..1)];
Min=min(f~(R)[], f(0), f(2*Pi));
Max=max(f~(R)[], f(0), f(2*Pi));


Unfortunately  method=branchandbound  option is not very reliable. The example below shows that both the minimum and maximum are found incorrectly, which is clearly shown by the graph:

plot(f, x=0..2*Pi, size=[1000,500], numpoints=1000);
Optimization:-Maximize(f, x=0..2*Pi, method = branchandbound);
Optimization:-Minimize(f, x=0..2*Pi, method = branchandbound);

DirectSearch  package successfully copes with the problem:

DirectSearch:-GlobalOptima(f, [x=0..2*Pi]);
DirectSearch:-GlobalOptima(f, [x=0..2*Pi], maximize);
 [-4.82606452700204, [x = 3.59004152503469], 191]
                              [3.40373784598079, [x = 4.94786409954761], 162]



@Bilawal  See another way in my answer. 

@Mariusz Iwaniuk  The results for  x>Pi/2  are erroneous:

F:=simplify(unapply(int(sqrt(sin(x)), x=0..T), T)) assuming T>0, T<=Pi;
F(Pi/4), F(Pi/2), F(3*Pi/4), F(Pi): 
# Examples

@nm  Sorry, I am wrong. I had in mind a syntax error for the numerical solution. I corrected the title.

@ahmeng  See help on  value(%)  and  emptysymbol(``)

@Lali_miani  In Maple, there is a rule: if at least one number is represented as a decimal fraction, for example  0.2  in your example, then Maple calculates the entire expression approximately (by default with 10 significant digits). If you want to have a symbolic representation, then instead of 0.2 you should write 1/5, that is  sin(1/5) . But then in the output Maple just writes the same thing, since sin(1/5)  can not be exactly written in a simpler form, for example, in radicals. But with this expression it is possible to work further, for example to calculate it approximately with some accuracy or to expand in a series, etc.



@Lali_miani  Unfortunately I'm not familiar with English-language books on Maple (English is not my native language). Maybe someone else will advise you a good book on Maple.

@Carl Love  But by this we have total 3 significant digits, rather than 3 digits after the decimal point:

interface(displayprecision= 3):

@vv  Thanks for this effective way (I did not know about it before). In addition to your answer here is a quote from the Help: "To compute  i^n mod m where  i  is an integer, it is undesirable to use this "obvious" syntax because the powering will be performed first over the integers (possibly resulting in a very large integer) before reduction modulo m. Rather, the inert operator  &^  should be used:  i &^ n mod m .  In the latter form, the powering will be performed intelligently by the mod operation. 

@Joe Riel  It's better. I'm used to using this syntax for lists, but for vectors it also works.

@ahlamsaqer  Write here your specific function.

@Adam Ledger  See an implementation of your formula in the update of my answer above.

@Adam Ledger  Because he has Maple 7 only. Here is the text of this code:

# # # # # # # # # # # # # # # # # # # # # # # # # # # # 
# Differences in the various "functions"
# # # # # # # # # # # # # # # # # # # # # # # # # # # # 

printf("Consider the folowing:\nf:=x^2+5*x:\ng:=x->x^2+5*x:\nh(x):=x^2+5*x:\n\n");
printf("f is of type %a\n",whattype(f));
printf("g is of type %a\n",whattype(g));
printf("h is of type %a\n",whattype(h));
printf("subs(x=2, f) gives %a\n", subs(x=2,f));
printf("f(hello) appends (whatever) to x - %a\n",f(hello));
f(hello);  #appends (whatever) to x;
printf("subs(x=2, f) gives %a\n", subs(x=2,f));
printf("...but f(2) gives %a\n", f(2));
printf("g(2) gives %a\n", g(2));
printf("Both h(2) and eval(h(2)) give %a\n", eval(h(2)));
printf("...but rather subs(x=2,h(x)) gives %a\n", subs(x=2,h(x)));


@brian bovril  My system is Windows 10, Maple 2017.3 32 bit. I have no more comments.

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