Where Does It Happen?
This problem is from Jason Holland.
Let x1, x2, x3, x4 be real numbers such that x2 – x1 = x3 – x2 = x4 – x3 = 1. Prove that the product x1x2x3x4 is never less than -1, but can equal -1. Find all lists (x1, x2, x3, x4) for which the product is equal to -1.
Please submit all solutions to mathpotw@acu.edu by Thursday, February 27th by 5:00 PM.
Solution to Where Does It Happen?
Correct solutions were submitted by Wyatt Witemeyer.
Consider a function
f(x) = (x-1)(x)(x+1)(x+2).Then the derivative of f(x) is
f ‘(x) = 4x3 + 6x2 – 2x – 2.If f ‘(x) = 0, then we obtain the equation
2x3 + 3x2 – x – 1 = 0.All local maxima and minima must occur at roots of this equation. The only potential rational roots are 1, -1, 1/2, and -1/2. A manual check shows that x = -1/2 works as a root, and it is the only one of the four that does. Using the fact that x = -1/2 is a root we can factor:
2x3 + 3x2 – x – 1 = (x+1/2)(2x2 + 2x – 2).We check to see that f ”(-1/2) is negative, so the function f has a local maximum at -1/2. Therefore the two places where f has local minimums are the roots of the quadratic equation
2x2 + 2x – 2.These roots are r1 = (-1 + SQRT(5))/2 and r2 = (-1 – SQRT(5))/2. A manual check verifies that f(r1) = f(r2) = -1. Therefore the two lists of four numbers are
((-3 + SQRT(5))/2, (-1 + SQRT(5))/2, (1 + SQRT(5))/2, (3 + SQRT(5))/2) and
((-3 – SQRT(5))/2, (-1 – SQRT(5))/2, (1 – SQRT(5))/2, (3 – SQRT(5))/2).Comments:
- The number (-1 + SQRT(5))/2 is a famous number, sometimes called the golden ratio. It is interesting to me that this number shows up here.
- The ideas in the previous problem of the week can be used to show that if we add 1 to the function f(x) = (x-1)(x)(x+1)(x+2), then we get a “perfect square”. Hence it easily follows that f(x) is no smaller than -1. But that does not help us find where this minimum occurs.
