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I have recently come across an article by Atara Shriki of the Technion - Israeli Institute of Technology - where she extended an engaging property of the graph of y = x³ introduced by R. Honsberger.

At an arbitrary point P on the graph of y = x³ draw the tangent line and mark its intersection with the graph. Denote the point Q. At Q draw another tangent and denote its intersection with the graph R. Let the area between PQ and the graph be A, that between QR and the graph B. Then

B = 16A.

Once you determined the equation of the tangent line, the proof becomes rather straightforward. Calculus students must be able to carry it out.

Honsberger made to observations that he did not pursue in his book. First, the result holds for any cubic parabola, not just y = x³. Second, he intimated that higher power function may have similarly engaging properties. Shriki answered the latter challenge.

First what she found is that, for any exponent n ≥ 3, the ratio x_Q/x_P is constant, independent of x_P, so that, e.g.,

x_R/x_Q = x_Q/x_P

The ratio, which I shall denote r, is solely a function of n. Further, she found that the ratios of the areas between the tangent and the curve is also constant: B/A = s, which again depends only on n.

Finally, Shriki has observed and subsequently proved that there is a relation between the two ratios, viz.,

r ⁿ⁺¹ = s.

It is interesting to investigate if Shriki's result extends to polynomials more general than y = xⁿ.

References

1. R. Honsberger, More Mathematical Morsels, MAA, New Math Library, 1991
2. A. Shriki, The area problem, Mathematical Spectrum, 39 (2006/2007) N 1, 27-31