Please help me with this question...
Find min value of $\displaystyle f(x)=(x-2006)(x-2007)(x-2008) (x-2009) $
Thanks..
Multiply it all out, take the derivative, set it equal to 0 and solve for x.$\displaystyle f(x)=(x-2006)(x-2007)(x-2008) (x-2009) $
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.
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.
"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.