January 15–January 21, 2008
Problem
A function f is continuous on a ≤ x ≤ b and differentiable on a < x < b. Furthermore, f(a) = a and f(b) = b. Show that there are two points x1, x2 such that a < x1 < x2 < b and 1/f '(x1) + 1/f '(x2) = 2.
Hint: recall the mean value theorem and the intermediate value theorem from calculus.
Solution
here.
List of solvers
We received ingenious solutions from Gary Raymond, Dustin Moody, Travis Willse, Ian Zemke, and Alan Jamison.
However, the problem was trickier than I suspected, and all the solutions relied on f ' being continuous. (Thanks, Chris, for catching this subtle detail!) It's hard to imagine a function that's continuous on [a,b], differentiable on (a,b), but whose derivative is NOT continuous. This can happen, though; there's a nice example with some discussion here.
Since there was no clear correct solution, I'll hold on to the prize, and then in some coming week have a two-part puzzle with two prizes.