00:01
Hello, we're going to sketch this graph here and we're going to find the domain, the x and y intercept, see if it's symmetric or not, and then we're going to make a sign chart for the first and second derivative, and then we're going to find the local extrema.
00:16
So let's get started.
00:19
So first, let's see what the domain is.
00:23
So the domain is going to be from negative infinity to infinity.
00:29
The reason is that for every single, point on the x -axis, x -axis, we see that it will give us a proper value on y.
00:45
So therefore, every single number on the x -exes will make sense.
00:49
Therefore, the domain is from negative infinity to infinity.
00:53
Now let's find the x -intercept.
01:00
So the x -intercept.
01:01
That means that when we have a graph and it hits the x intercept, that is going to be our x intercept.
01:13
And if you notice, the x intercept happens when y equals 0.
01:18
So f of x equals 0.
01:22
So what we can do is that we can set this equation, this equation here, equal to 0.
01:33
So we have x 5 3rd minus 5 5 3.
01:40
X two thirds equal zero and that is the same as x five -thirds minus or i made equal to five x two -thirds and so we see that these are the same as the cube root of x to the power of five equals to the cube root of x to the power of two and since we both sides have a cube root we can cube on both sides.
02:21
So what we're going to have is we're going to have this three cancel with that three, this three, cancel with that three.
02:30
We're going to end up having x to the power of five equals x to the power of two.
02:39
So what value of x can we think of that when we raise it to the fifth power is the same as when we raise to the second power.
02:49
Well, first of all, we can see that it cannot be any negative number.
02:54
So let's say a is our solution, that it cannot be a negative number.
02:59
Or it cannot be a negative number because this is an odd power.
03:05
So if it's an odd power, then when we plug in a negative number, we will have a negative number, while this side is an even power.
03:15
So any negative number, when we square it, we get a positive number.
03:21
And so immediately we can see that, oh, so the solution has to be equal or larger than zero.
03:28
So let's see what zero is going to give us.
03:32
So zero, we know that zero to any power will give a zero.
03:38
And so therefore, zero to the power of five is also going to equal to zero to the power of two.
03:44
And so the x intercept is going to be at zero and then we can also see that huh what happens if we cancel this out so we can divide on both sides by x to the power of two so we divide both sides by x to the power of two this will give us 1 over here.
04:18
And so therefore, this side will have x to the power of 3.
04:24
So what number do we put in here to give us 1? well, if we simply take the cube root of both sides, we see that the cube root of 1 is 1.
04:35
It's not negative 1 because if we take the q root of negative 1, we're going to get negative 1.
04:40
And so the x intercept is going to be at x zero and x equals one.
04:53
And so next up, we are going to have the y intercept.
05:04
So the y intercept happens when the graph touches y.
05:09
That happens when x equals zero.
05:14
So what we can do is because we know that x is, because we know that x is, 0.
05:21
So we can simply plug in 0 to every single x in the equation.
05:29
So we have g of x equals to x 5 thirds minus 5x 2 thirds.
05:40
And so if x was 0, then that'll be 0, right? because 0 to the 5 thirds equals 0, minus 5 times 0 to the power of 2 thirds.
05:51
That will also give us 0.
05:54
And so 0 times 5 will give us 0 and 0 will be 0.
06:00
So 0 minus 0, g of x is going to be 0 when x equals 0.
06:09
So that means the graph crosses the origin at 0.
06:18
So so far, let's make a little table here.
06:23
So intercept at x equals 0 x equals 1 and y intercept happens at origin and so basically these two they're going to be the same thing so next up we are going to see what the extrema is going to be so the extrema happens when the derivative of the graph equals zero.
07:02
Because if you think about it, when the graph is like that, this is going to be the tangent line to the graph, right? that's going to be the slope of the graph.
07:11
If it's zero, then the graph is not changing, so it's going to move on to the other direction.
07:18
So what we can do is we can find the derivative of the graph, and then we can set it equal to zero.
07:24
So let's find the derivative of g of x is going to be d.
07:30
D of x of x of x 5 thirds and then minus 5x 2 thirds.
07:44
So using the power rule, we can say that the derivative of the first part is going to be 5 thirds of x 5 thirds minus 1, minus 5 times 2 thirds.
08:05
X of two -thirds minus one.
08:10
So that is going to give us five -thirds.
08:16
So five -thirds minus one is going to be the same as five minus three divided by three.
08:23
So that's going to give us two -thirds.
08:26
So that's going to give us two -thirds minus ten -thirds times x to two -minous two -minous two -minus two -minus three, divided by 3, that's going to give us negative 1 3rd.
08:39
So that's going to be negative 1 3rd.
08:44
It's the same as 1 divided by x to the power of 1 3rd.
08:56
Sorry.
08:58
So that is going to be our derivative function.
09:04
So what we can do now is we can set that equal to 0 because that is where our extremer is going to be.
09:11
So then we can see that this expression, same as 5 thirds, x, 2 thirds, equals to 10 divided by 3 times 1 divided by the cube root of x.
09:31
So what we can do over here is that we can solve for x.
09:35
So we see that this and that they cancel out, and this and that they cancel out.
09:41
What we're going to have left is going to be 2.
09:44
Right there.
09:47
And so when we multiply both sides by x to the power of one -thirds, we can simply cancel this part out.
09:58
And so when we multiply powers, what we're going to do is we put the same base there, and then we're going to put one -third plus two -thirds, which is going to be one hole.
10:14
So x equals two.
10:16
So our extrema is going to be at x equals 2...