F (x) = - x^ (4) - 9x^ (3) - 24x^ (2) - 16x what is the relative maxium and minimum?

We'll need to find the 1st and 2nd derivatives of F (x) to answer that question.

F ' (x) = - 4x^3 - 27x^2 - 48x - 16 You must set this = to 0 and solve for the

roots (which we call "critical values).

F " (x) = - 12x^2 - 54x - 48

Now suppose you've found the 3 critical values. We use the 2nd derivative to determine which of these is associated with a max or min of the function F (x).

Just supposing that 4 were a critical value, we ask whether or not we have a max or min of F (x) there:

F " (x) = - 12x^2 - 54x - 48 becomes F " (4) = - 12 (4) ^2 - 54 (4)

= - 192 - 216

Because F " (4) is negative, the graph of the given

function opens down at x=4, and so we have a

relative max there. (Remember that "4" is only

an example, and that you must find all three

critical values and then test each one in F " (x).