You may know that the expressions inside the csgns are real, but do you know wheher they are positive or negative?

How can you expect anyone to answer that when you've given no formulas, no code, and no reference material?

@Scot Gould Why do you refer to 2*x + 7 = x^2 + 4*x + 4 as the original equation? The original equation is sqrt(2*x+7) - x - 2 = 0.

@janhardo You can't divide the interval by n-1 if n=1. Acer's `if` is there to treat n=1 as a special case.

@mmcdara The second solution that I gave in my Answer doesn't require DataFrame, will work in Maple 17, and places the row labels on the right. So, there'll be no plot needed.

@Joe Riel According to Stefan's writing above

• [The bound variable] is quietly implicitly declared local if one wasn't declared

To me, that adequately explains why the protected problem no longer occurs. But it doesn't explain the absence of the assertlevel problem.

@janhardo If e1 and e2 are expressions and c is a condition that evaluates to true or false, then

if c then e1 else e2 fi

is often abbreviated

`if`(c, e1, e2)

Transposition is an involution (f(f(x))=x); it's not idempotent (f(f(x)) = f(x)). A linear idempotent function is called a projection.

@janhardo Yes.

Let me know if you need any mathematical hints to compute the x-interval needed to obtain a particular length of tangent line.

Why do you want the extrema of the spline rather than the extrema of the original data?

@janhardo I suggest making the x-ranges of the tangent lines (tangency point) +- (b-a)/2/n where n is the number of tangency points. Currently, you have n hard-coded as 6. If you want to improve this procedure, my next suggested step is to make this 6 a variable.

@janhardo Since a and b are now parameters of the procedure, they can't be locals. You simply need to remove them from the local list. While you're at it, you might as well add xaxis to the locals.

Then if you run the procedure, it'll work, but you'll see that the tangent lines are too long. Can you correct that?

@ Please read the help pages ?assume, ?assuming, and ?assuming,details, paying particular attention to the additionally command and the additionally subclause of assuming. Then, let me know if you have further questions.

@janhardo There are two ways to make comments in Maple code. The first style is #, which causes all input to the end of the line to be a comment. I use this style for explanatory comments. The other style is (* ... *), which causes all input, including line breaks, between (* and *) to be ignored. I use this style for commenting out code.

@ Like I said in my Answer below, which you apparently acknowledged, the algorithm is called branch-and-bound. How did you miss that?