00:01
We're in search of the regions of concavity and inflection points of a function.
00:05
We do know that to look at that, we need to get to the second derivative.
00:11
Our first derivative, using the power rule, would be negative 4x cubed minus 6x squared plus 24x.
00:22
And our second derivative, again, using the power rule, would be negative 12x squared minus 12x, plus 24.
00:35
The critical points, or should be possible inflection points on this, which i'm going to call pips, possible inflection points would be where the second derivative is equal to zero or where it's non -differentiable.
00:48
It is never non -differentiable, though, because it's a polynomial function.
00:52
So let's just set it equal to zero.
00:59
Now, i can factor out a negative 12, which will give me negative 12, parentheses x squared plus x minus two, which factors to x plus two times x minus one? okay, so that equals zero.
01:23
So if that equals zero, that either means the x plus two is zero or the x minus one is so we have possible inflection points when x is negative 2 and when x is positive 1.
01:39
If we do a number line test for points around them, and remember, we're filling this into the second derivative that we had found earlier up here...