@tomleslie Thank you very much. A small question. Why I get error when I use nops(evalf(sol)) ?

@Carl Love How can I get the exact of the numbers a, b, c?
By hand, I can prove that, the line Delta passing the point DD and perpendicular to the plane (ABC). And then,

maxS = Distance(A, DD)+Distance(B, DD)+Distance(C, DD);

My code get 

[9.37769297305868754, [a = .801783725149603, b = .534522484467394, c = .267261242390448]]

@Joe Riel Thank you very much.

@mmcdara Because I choose polynomial degree 10, then when tends to plus or minus infinity, f(x) always tend to infinity. It didn't fit with figure. 
I think, we need to choose a function that is suitable for the drawing. 

@tomleslie I like your answer. It closes with my picture. I copied your code to LaTeX and used piecewise. It looks good (except that the dots are maybe not exactly at the positions of the local extrema). Can you repair positions of the local extrema?

@mmcdara With the given figure, When x  tends towards  +infinity, then f(x) tends towards  +infinity, too?

@tomleslie Thank you very much.

@vv Thank you very much.

@vv How about real in command? limit(sqrt(x^2-3*x+2), x = 2, real)

@vv Can you post your code?

@acer How can I get the arrow of each axis (ox and oy)? For exmaple

@vv How about ^+ in the line?

solve( (P-H)^+ . (B-A), t ):

@Kitonum Thank for your code. I tried
 

restart:
k := int(cos(3*x)^21, x):
algsubs(cos(3*x)^2 = 1-sin(3*x)^2, %):
a := factor(%):
b := expand(algsubs(sin(3*x) = t, a)):
algsubs(t = sin(3*x), %)

@acer Thank you very much.

@Kitonum How about tan(x)^7?
 

restart; 
`assuming`([int(tan(x)^n, x)], [n::posint]);
 eval(%, n = 9); 
expand(simplify(value(%))); 
algsubs(tan(x)^2+1 = 1/cos(x)^2, %)

I could not get the form

tan(x)^8/8 - tan(x)^6/6 + tan(x)^4/4 - tan(x)^2/2 - ln(cos(x))