I can't for the life of me figure this one out all the way.(adsbygoogle = window.adsbygoogle || []).push({});

Suppose f : R -> R is differentiable, and consider g(x) = f(f(x)). Show that if g is monotone decreasing, then g must be constant.

Here's what I've done so far (I'd hesitate to call it "progress"):

By the chain rule, g'(x) = f'(f(x)) f'(x). Suppose that g is strictly decreasing so that f'(f(x)) f'(x) < 0. One of the factors is positive and the other is negative. Since, by Darboux's theorem, f'(x) has the intermediate value property, there exists a q between x and f(x) such that f'(q) = 0. Then g'(q) = f'(f(q)) f'(q) = 0, a contradiction. Therefore g is not strictly decreasing.

I've investigated the fixed points of f and found lots of interesting facts, but none of them seems to lead anywhere on this problem.

Any ideas or suggestions? Thanks a lot.

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# A question involving self-composition.

Loading...

Similar Threads - question involving self | Date |
---|---|

B Function rules question | Saturday at 9:49 AM |

I Help please with biocalculus question involving differentiation | Nov 4, 2017 |

Question involving Levi-Civita symbol | Mar 28, 2013 |

Questions involving differentials (again) | Sep 4, 2012 |

Question on complex integration involving Bessel Function and essential Singularities | May 8, 2012 |

**Physics Forums - The Fusion of Science and Community**