00:03
Okay, so i want to start by filling in the coordinates of critical points that are max, min, or neither.
00:16
Okay, so i'm going to find the derivative of this function using power rule.
00:22
It's going to be 24x cubed plus 48x squared.
00:29
I'm going to set that equal to zero and factor out a 24.
00:34
X squared and that is going to give me x plus two so those critical numbers are zero and negative two i'm going to put them on a number line and choose some test points to the left and the right and just plug in um negative three into my first two derivative.
01:16
So when i plug in negative 3, i'm going to get out a negative value.
01:23
Negative 1 is going to give me, i'll just jot that down so you can see.
01:29
K prime of negative 3 is negative 216.
01:34
K prime of negative 1 is 24.
01:37
So that is positive.
01:40
And then k prime of positive 1 is positive 1 is.
01:46
72 so that's going to stay positive so i am going to have let's see because my graph is going to go down right that's decreasing and then up and then it's going to stay up so i'm going to have a relative min and that relative min is going to have an x coordinate of negative 2 and i'm going to plug that into now my original function and that is going to give me negative 32 so i'm going to have a relative min at negative 2 negative 32 and there will be no relative maximum points now increasing and decreasing.
02:47
So my graph is going to start off decreasing, and it's going to decrease from negative infinity until i get to that relative minge is negative 2, at which point it is going to be increasing, so it's going to increase from negative 2, and then it's not going to stop increasing.
03:12
So you want to choose for this one, choice a, but do the increasing first and then the decreasing.
03:26
See, we want points of inflection.
03:29
So we're going to need the second derivative.
03:35
24 times 3 is, that's 72x squared, plus 48 times 2 is 96x...