How do use the first derivative test to determine the local extrema #f(x) = 3x^5 - 20x^3#?
You determine the crititcal points of the function and check to see if any of these points are local minima or local maxima.
Basically, the first derivative test allows you to look for a function's critical points and check to see whether or not these points are local extrema by looking if the sign of the first derivative changes around them.
So, the first thing you need to do is determine the first derivative of
To find the function's critical points, make the first derivative equal to zero and solve for
This equation has three solutions,
Since your function has no domain restrictions, all these points will be critical points.
Now, in order for the points to be local extrema, the function must go from increasing, i.e. a positive
Since you have 3 critical points, you're going to look at 4 intervals. Select a value from each itnerval to determine the sign of the first derivative on that interval
#(-oo, -2)# #(-2, 0)# #(0, 2)# #(2, 0)#
So, you know that the first derivative changes sign, i.e. it goes from positive to negative, around
On the other hand, the first derivative does not change sign around point
Notice that the first derivative changes sign again around point
To get the actual points that match these critical points, evaluate the original functions in
Therefore, the function