# Thread: Find the minimum value of..

1. ## Find the minimum value of..

Find min value of $f(x)=(x-2006)(x-2007)(x-2008) (x-2009)$

Thanks..

2. ## Re: Find the minimum value of..

$f(x)=(x-2006)(x-2007)(x-2008) (x-2009)$
Multiply it all out, take the derivative, set it equal to 0 and solve for x.

Take the 2nd derivative and evaluate it at each of these points.

Doing this you find 3 stationary points. 2 are minima, 1 is a local maxima.

You can work out the details.

3. ## Re: Find the minimum value of..

Since this equation is a 4th order polynomial, the derivative is a cubic which will be difficult to factor. One way to make this easier is to shift the values to a range that is easier to work. For example if you let w = x-2007.5, then finding the min of (w-3/2)(w-1/2)(w+1/2)(w+3/2) = w^4 -5/2w^2+9/16 is much easier. Solve that, then add 2007.5 to get the answer to this problem.

4. ## Re: Find the minimum value of..

"This problem" asks for the maximum value of f. It is not necessary to "add 2007.5". Evaluating w^4- (5/2)w^2+ 9/16 at the value of w that gives a maximum will give the maximum value of f. Adding 2007.5 to that value of w will give the corresponding value of x.

5. ## Re: Find the minimum value of..

You're right - I was thinking that the value of x was needed, but it's not. Thanks.

6. ## Re: Find the minimum value of..

Originally Posted by Deci
Find min value of $f(x)=(x-2006)(x-2007)(x-2008) (x-2009)$