2009 Indonesia MO Problems/Problem 6

Problem

Find the lowest possible values from the function $f(x) = x^{2008} - 2x^{2007} + 3x^{2006} - 4x^{2005} + 5x^{2004} - \cdots - 2006x^3 + 2007x^2 - 2008x + 2009$ for any real numbers $x$ .

Solution

We start the search for potential minimums by taking the derivative of $f(x)$ and setting it to equal 0. $f'(x) = 2008x^{2007} - 4014x^{2006} + 6018x^{2005} - \cdots - 6018x^2 + 4014x - 2008 = 0$ Notice from the symmetry of the function that $f'(1) = 0$ . This makes $f(1)$ a possible minimum of the function.

To check, we find that $\begin{align*} f(x) &= (x-1)(x^{2007} - x^{2006} + 2x^{2005} - 2x^{2004} + \cdots + 1004x - 1004) + 1005 \\ &= (x-1)^2(x^{2006} + 2x^{2004} + 3x^{2002} + \cdots + 1004) + 1005 \end{align*}$ From the Trivial Inequality, $(x-1)^2 \ge 0$ and $x^{2006} + 2x^{2004} + 3x^{2002} + \cdots + 1004 > 0$ , so $(x-1)^2(x^{2006} + 2x^{2004} + 3x^{2002} + \cdots + 1004) \ge 0$ . Thus, the minimum of $f(x)$ is $\boxed{1005}$ , obtained when $x = 1$ .

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

2009 Indonesia MO (Problems)
Preceded by Problem 5	1 • 2 • 3 • 4 • 5 • 6 • 7 • 8	Followed by Problem 7
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2009 Indonesia MO Problems/Problem 6

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