4-3-29
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Taking the first derivative and setting it equal to 0 and
factoring, we see that we get -1 zero and one as our critical
points.
So now if we do our whether it's increasing or decreasing by
putting in values, say choose something to the right of 1-10,
five, 100, we're going to get a positive times a negative times
a positive.
So those are going to all be negative to the right of one at
1.
Due to its odd multiplicity, it's going to change to
positive.
At 0.
Due to its odd multiplicity, it's going to change to
negative.
At -1.
Due to its odd multiplicity, it's going to turn back to
positive.
So the original function from negative Infinity to -1 is going
to be increasing, from -1 to 0 it's going to be decreasing,
from zero to one it's going to be increasing, and from one to
Infinity it's going to be decreasing.
So now if we put an H of -1, H of 0 and H of one, we get 1/3
for H of -1 and H of one.
So the fact that those both are the same values tells me that
those are absolute Maxis.
The zero at 0 has got to be a local min because the original
function is going to keep going negative in both directions.
So there is no absolute minimum, but there is a local minimum at
00.
So that's a local min.