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During the summer of 1995 the authors engaged in an undergraduate research program that investigated various conjectures about orthogonal polynomials. While exploring the grahic capabilities of Mathematica, we generated Figure 1, which shows, on a single set of axes, the fifth-degree Jacobi polynomials, for beta = 0.9 and alpha taking the values 0, 1, 2, ..., 6. We observed that the curves in the figure appear to intersect at extrema, but our advisors were skeptical about whether this actually happened. We used Mathematica to confirm that it did.
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