00:01
Okay, so in this video we're asked to answer a bunch of questions about the function fx equals 2 x cubed minus 18 x squared plus 48 x minus 7.
00:16
So let's just handle them one by one.
00:19
Part a just says find the derivative f prime of x.
00:24
And here you just have to remember the rules for the derivatives, right? so we have an x cubed, x cubed the three comes down.
00:29
You get 3x squared times the 2 is 6 so 6 6 x squared the x squared derivative is 2x so you have minus 18 times 2x so minus 36 x plus 48x just got the 48 and derivative the constant term is 0 right so this is the derivative of the function part b asks find the critical, find the critical points.
01:03
And we just have to recall that the critical points are the solution of f prime equals zero.
01:08
So we have to solve 6x squared minus 36x plus 48 equals 0.
01:16
And let's just make our life easier and divide everything by 6.
01:19
If i divide both sides by 6, i get x squared minus 6x plus 8 is equal to 0 divided.
01:27
By 6, which is 0.
01:29
This is easier to solve, and this is a quadratic, just a quadratic polynomial, so we use the quadratic formula.
01:37
So minus b, 6 plus minus square root b squared, 36 minus 4 times a, which is 1 times c, which is 8, 4 times 8, 4 times 8.
01:47
4 times 1 times 8.
01:49
If you're like this, and then make an extra strip, divided by 2 times a, which is just 2.
01:54
So you get x is 6 plus minus 30, this is 32, 36 minus 32 is 4, root of 4 is 2.
02:02
So 6 plus minus 2 over 2.
02:05
So we get the two solutions.
02:08
X and 6 plus 2 is 8 over 2 is 4.
02:12
6 minus 2 is 4 over 2 is 2.
02:18
All right.
02:18
So these are our two critical points.
02:21
So critical points are x equals 2 and x equals 2 and x for.
02:30
Okay, now, part c, i'm going to combine part c, d, and e together.
02:36
It's not, they don't have letters, but okay, part c, let me just answer everything together.
02:43
Find intervals where f increases, where f decreases, and then the minima and the maxima.
02:55
Right, and i'm going to do all of it in a table.
02:57
All right, so we've already seen what the important points are 2 and 4.
03:01
So i'm going to have minus infinity.
03:03
Then between minus infinity, i'm going to have the point 2 in here.
03:12
Then i'm going to have 4.
03:13
And then this will be plus infinity, just at the end...