00:01
We now determine the intervals of increase or decrease and also determine the intervals of concave up and concave down for this function y equals x -cube plus 3x squared plus 3x plus 3.
00:13
So to determine the intervals of increase or decrease, we basically have to prepare the first derivative sign chart.
00:20
And to determine the concave up or concave down intervals, we have to prepare the second derivative sign chart and determine the intervals of concave up or concave down.
00:30
So first let's start by preparing the first derivative to sign chart.
00:35
With this, we can determine the intervals of increase or intervals of degrees.
00:40
So first, let me write down this function in function notation as f of x.
00:44
And this equals x cube plus 3x squared plus 3x plus 3.
00:52
Now we find the derivative of this function, that is f prime of x.
00:57
And this equals the derivative of x cube using the power rule.
01:01
Is 3x squared and then the derivative of 3x squared is 3 times of derivative of x squared is 2x using the power rule so it is 3 times of 2x plus the derivative of 3x is we put the constant 3 the derivative of x is 1 and then the derivative of 3 which is a constant is 0 so therefore we get f prime of x which is the derivative of f of x this equals 3x squared plus 3 times 2x is 6x plus 3.
01:37
So first we have to find the critical point of this function so that we can replace the critical points to prepare the first derivative 2 sign chart.
01:46
We find the critical points by setting up this equation to 0, that is f prime of x equal to 0 and solving for x.
01:55
So replace the first derivative to f prime of x by 0 and solve for x.
02:00
We get 0 equals 3x squared plus 6x plus 3.
02:05
Now on the right side i can factor 3.
02:09
And when i do that, i get x squared plus 2x plus 1.
02:14
And in fact, this can be written as 0 equals 3 times of x plus 1 quantity squared.
02:23
We write down this quadratic expression x squared plus 2x plus 1 as a binomial square as x plus 1.
02:30
Quantity square.
02:31
Dividing both size by 3, we get 0 equals x plus 1 20 square.
02:39
Now if we take square root on both sides and solving for x, we get the value of x equals negative 1.
02:46
So this is the critical point of the function.
02:52
Now let's see how to determine the first derivative sign chart.
02:56
So for that i should put line number, the number line and mark the critical point.
03:03
In this case, we have only one critical point.
03:06
So i mark negative 1.
03:08
And basically, we need to understand the sign of this first derivative 2 to the left and to the right of the critical point.
03:16
So for that, we take some sample points, plug it into the first 2 derivative 2 and understand its sign.
03:22
So if you look at this first derivative 2, which is, i have written this as 3 times of x plus 1 quantity square.
03:31
This is always going to be positive irrespective of the value of x whether we plug in any values to the left of negative one or to the right of negative one whether it is positive values or negative values is always going to be greater than zero so therefore we mark this sign chart as like this that is it is positive to the left of negative one as plus positive to the left to the right of negative one and this means we have out two intervals, both are increasing intervals.
04:04
Whenever the first derivative is positive, we see that it is an increasing interval.
04:09
And this is also increasing interval.
04:12
And so this interval is negative infinity to negative 1.
04:17
And then we union this with the second interval.
04:21
That is negative 1 to infinity.
04:24
So therefore, we see that the function has increasing on negative 1 to negative 1.
04:31
Union negative 1 to infinity...