Problem #7

ABSTRACT

The function F(x) = (1/2-x) (1-x²)^1/2+x(1+(1-(1/2+x)²)^1/2) has a maximum at about x = .343771, where it attains the value of approximately .674981. This value is the root of an irreducible polynomial of tenth degree over the integers; the problem is to find this polynomial. The obvious way of proceeding is as follows:(1) Differentiate F(x), set it equal to zero, and clear radicals. The result is a tenth degree polynomial P(x) over the integers which has a root at about x = .343771.

REFERENCES

• 1: ALTRAN User's Manual (Vol. I), W. S. Brown, available from Bell Telephone Laboratories, Inc., Murray Hill, New Jersey 07974.
• 2: R. L. Graham, The Largest Small Hexagon (to appear).