Maxwell-Area Construction Problem

Question

I am solving a problem which asks to find the equation of a horizontal line which crosses the graph of $$y=x^3-3x+1$$ at three distinct points, such that the two areas bounded by such curves are equal.

I am creating a program for this problem. What I did is to solve for the roots or the points of intersection of the 2 curves (say $a,b,c$) analytically. Suppose the horizontal line takes the form $y=y_p$. The roots are functions of $y_p$ only. Then I used composite Simpson's 3/8 rule to compute for the areas which should be numerically equal. I got $y=1$ as the final answer.

My problem is, as an alternative since analytical formulas for cubic equations is very tedious to code, is it possible to solve for the roots by Newton's method even if $y_p$ is still unknown?

Answer 1

Hint The graph of any cubic function $$f(x) := A x^3 + B x^2 + C x + D ,$$ $a \neq 0$, is symmetric about its unique inflection point: We can show this by solving $f''(x) = 0$ to show that the unique inflection point is $(s, f(s))$, where $s := -\frac{B}{3 A}$, and then showing that $x \mapsto f(x - s) - f(s)$ is an odd function.

Additional hint So, by symmetry, the horizontal line must pass through the inflection point, and hence it is the equation with $y = f(s)$.

Note that simply by appealing to symmetry we can avoid actually computing the intersection points or the areas of the bounded regions, which like the question indicates, is awfully unpleasant, on account of the difficult of extracting roots of general cubics.

Answer 2

HINT:

Advantage of differentiation is obvious. The derivative has roots $ x = \pm 1 $ for extremal points. It appears the arbitrary constant in $ y = x^3 - 3 x + C $ is much easier to handle, where the cubic is displaced arbitrarily parallel to $y-$axis.

Answer 3

the symmetric of function about original is there in all terms of equation except the $1$, so if we down the equation by one, the function will become symmetric. $$y=x^3-3x$$

now we can conclude the $y=1$ is the equation of horizontal line