0:00:00.200,0:00:01.033
- [Instructor] We're told to consider

0:00:01.033,0:00:02.830
the curve given by the equation.

0:00:02.830,0:00:04.070
They give this equation.

0:00:04.070,0:00:07.210
It can be shown that the[br]derivative of y with respect to x

0:00:07.210,0:00:09.610
is equal to this expression,[br]and you could figure that out

0:00:09.610,0:00:11.180
with just some implicit differentiation

0:00:11.180,0:00:14.160
and then solving for the[br]derivative of y with respect to x.

0:00:14.160,0:00:15.900
We've done that in other videos.

0:00:15.900,0:00:18.890
Write the equation of the horizontal line

0:00:18.890,0:00:22.960
that is tangent to the curve[br]and is above the x-axis.

0:00:22.960,0:00:25.960
Pause this video, and see[br]if you can have a go at it.

0:00:25.960,0:00:28.300
So let's just make sure[br]we're visualizing this right.

0:00:28.300,0:00:30.250
So let me just draw a[br]quick and dirty diagram.

0:00:30.250,0:00:33.800
If that's my y-axis, this is my x-axis.

0:00:33.800,0:00:35.860
I don't know exactly what[br]that curve looks like,

0:00:35.860,0:00:38.350
but imagine you have some type of a curve

0:00:38.350,0:00:41.630
that looks something like this.

0:00:41.630,0:00:44.130
Well, there would be two tangent lines

0:00:44.130,0:00:47.560
that are horizontal based[br]on how I've drawn it.

0:00:47.560,0:00:49.280
One might be right over there,

0:00:49.280,0:00:50.660
so it might be like there.

0:00:50.660,0:00:54.120
And then another one might[br]be maybe right over here.

0:00:54.120,0:00:56.010
And they want the equation[br]of the horizontal line

0:00:56.010,0:00:59.340
that is tangent to the curve[br]and is above the x-axis.

0:00:59.340,0:01:00.180
So what do we know?

0:01:00.180,0:01:05.180
What is true if this[br]tangent line is horizontal?

0:01:05.570,0:01:08.550
Well, that tells us that, at this point,

0:01:08.550,0:01:11.500
dy/dx is equal to zero.

0:01:11.500,0:01:13.890
In fact, that would be true[br]at both of these points.

0:01:13.890,0:01:15.940
And we know what dy/dx is.

0:01:15.940,0:01:19.190
We know that the derivative[br]of y with respect to x

0:01:19.190,0:01:22.340
is equal to negative[br]two times x plus three

0:01:22.340,0:01:26.280
over four y to the third[br]power for any x and y.

0:01:26.280,0:01:28.860
And so when will this equal zero?

0:01:28.860,0:01:30.970
Well, it's going to equal[br]zero when our numerator

0:01:30.970,0:01:34.060
is equal to zero and[br]our denominator isn't.

0:01:34.060,0:01:36.450
So when is our numerator going to be zero?

0:01:36.450,0:01:38.520
When x is equal to negative three.

0:01:38.520,0:01:42.730
So when x is equal to negative three,

0:01:42.730,0:01:45.850
the derivative is equal to zero.

0:01:45.850,0:01:48.870
So what is going to be[br]the corresponding y value

0:01:48.870,0:01:50.780
when x is equal to negative three?

0:01:50.780,0:01:52.650
And, if we know that, well, this equation

0:01:52.650,0:01:54.600
is just going to be y[br]is equal to something.

0:01:54.600,0:01:56.480
It's going to be that y value.

0:01:56.480,0:01:57.550
Well, to figure that out,

0:01:57.550,0:01:59.480
we just take this x equals negative three,

0:01:59.480,0:02:01.600
substitute it back into[br]our original equation,

0:02:01.600,0:02:03.230
and then solve for y.

0:02:03.230,0:02:04.290
So let's do that.

0:02:04.290,0:02:08.080
So it's going to be negative three squared

0:02:08.080,0:02:10.229
plus y to the fourth

0:02:10.229,0:02:14.180
plus six times negative[br]three is equal to seven.

0:02:14.180,0:02:15.570
This is nine.

0:02:15.570,0:02:17.750
This is negative 18.

0:02:17.750,0:02:19.990
And so we're going to get y to the fourth

0:02:19.990,0:02:23.470
minus nine is equal to seven,

0:02:23.470,0:02:25.380
or, adding nine to both sides,

0:02:25.380,0:02:28.910
we get y to the fourth[br]power is equal to 16.

0:02:28.910,0:02:30.930
And this would tell us that y is going

0:02:30.930,0:02:33.400
to be equal to plus or minus two.

0:02:33.400,0:02:36.100
Well, there would be then[br]two horizontal lines.

0:02:36.100,0:02:37.880
One would be y is equal to two.

0:02:37.880,0:02:40.810
The other is y is equal to negative two.

0:02:40.810,0:02:43.330
But they want us, the equation[br]of the horizontal line

0:02:43.330,0:02:47.550
that is tangent to the curve[br]and is above the x-axis,

0:02:47.550,0:02:51.130
so only this one is going[br]to be above the x-axis.

0:02:51.130,0:02:52.000
And we're done.

0:02:52.000,0:02:54.033
It's going to be y is equal to two.