how to find the absolute extrema of the function at a close interval?

Question

f(x) = 2(3-x) , [ -1, 2 ]

Answer 1

Absolute extrema are your maximum and minimum values for a give function over a range. In your example, this isn't too hard since the function is a line. All you really need to do is to plug in your endpoints and see what the values are:

f(x) = 2(3 - x)

f(-1) = 2(3 - (-1))
f(-1) = 2(4)
f(-1) = 8

f(2) = 2(3 - 2)
f(2) = 2(1)
f(2) = 2

So the absolute maximum for your given function over the range is at 8 (when x = -1) and the absolute minimum is 2 (when x = 2). Note that this only worked out since your equation is a line - if you have a higher order function (i.e. something with a squared term or higher) then you'd have to take derivatives and do some analysis with those.