How do you find all relative extrema of the function #f(x)= -x^3 -6x^2-9x-2#?
1 Answer
Use the first derivative test and check for sign changes of
Explanation:
For a given function, relative extrema, or local maxima and minima, can be determined by using the first derivative test, which allows you to check for any sign changes of
For a critical point to be local extrema, the function must go from increasing, i.e. positive
So, start by determining the first derivative of
To determine the function's critical points, make
This is equivalent to
Since no domain restrictions are given for your function, both solutions will be critical points.
Now check to see if the first derivative changes sign around these points. Since you're dealing with two critical points, you're going to have to look at 3 intervals.
Select a value fro meach of these intervals and note the sign of
#(-oo,-3)#
#(-3,-1)#
#(-1, oo)#
The first derivative changes sign twice. It goes from being negative to being positive around
On the other hand, it goes from being positive to being negative around point
This is equivalent to having a function that goes from decreasing to increasing (think of a valley) around point
To get the actual points at which the function has the local minium and maximum, evaluate
and
Therefore, the function
graph{-x^3 - 6x^2 - 9x - 2 [-10, 10, -5, 5]}
