H2 Math : Algebra

• Mad Hat

  2 Mar 13, 20:24

  I'm still curious about the minimum value of the function:

  f(x) = (x-2006)(x-2007)(x-2008 )(x-2009).

  Wen Shih, what is the best way to handle this? Probably there is some quick and clever way to see the minimum value of the function but I can't find it?

  The actual question is from playfulgirl here: http://www.flickr.com/photos/mabuthaodang/8515758304/in/photostream

  Some things are obvious, like the min. value is at least 0. Also the function clearly can take negative values, e.g. choose x such that the first term is positive and the other three negative. E.g. x = 2006.5.

  Also, the 2006, 2007, 2008, 2009 are not important. The question is equivalent to finding the minimum value of: f(x) = x(x-1)(x-2)(x-3). because the difference is just a horizontal translation of the graph along the x-axis.

  We can do the question the long way, e.g. expand the function, differentiate, and set to 0, but I'm curious what is the short way.

• wee_ws

  3 Mar 13, 07:32

  Hi,

    The minimum value of

    f(x) = (x-2006)(x-2007)(x-2008)(x-2009)

    is the same as the minimum value of f(x) = (x-1)(x-2)(x-3)(x-4).

    I considered the graphing approach (due to its ease of visualising any pattern) and observed that the minimum value of the curve

    y = (x-1)(x-2)(x-3)(x-4)

    occurred at one point where the line

    y = x-1

    intersects the curve y = (x-1)(x-2)(x-3).

    Solving, we obtain x = (5/2) +/- (sqrt(5)/2).

    Take x = (5/2) - (sqrt(5)/2), since the curve

    y = (x-1)(x-2)(x-3)(x-4)

    is symmetrical about the line x = 5/2 and we arrive (after some algebra) at the minimum value of -1.

  P.S. I think that the olympiad question setter typically develops interesting questions (like tha one above) that rely on some mathematical pattern or special mathematical behaviour.

    Thanks.

  Cheers,
  Wen Shih
  wenshih.wordpress.com

• Mad Hat

  3 Mar 13, 15:51

  Wen Shih thanks. So it's an Olympiad question? I was trying to find a "short way" with the question but maybe there's none! If I use the online GC, it also does show two minimum points, both of them -1.

• wee_ws

  3 Mar 13, 21:16

  Hi,

    Looks like one and tedious to solve if we were to use traditional methods :P

  Cheers,
  Wen Shih
  wenshih.wordpress.com