00:01
For this problem, we want to find the points xy, where the tangent line to the curve, y equals 1 plus 60x cubed, minus 2x to the 5th, has the largest slope.
00:12
Now, the first thing you have to do is to determine the objective function.
00:17
Now, the objective function is the function that is going to be optimized.
00:22
And based on this, we are trying to maximize the slope of the tangent line.
00:27
So let's call this our s.
00:30
Since the slope of the tangent line is determined by the first derivative of the given curve, then this is equal to y prime of x.
00:39
And because y prime of x is equal to 180x squared minus 10x to the fourth power, then our objective function is s equal to 180x squared minus 10x to the fourth power.
00:57
So now we want to find.
00:59
Find the critical points of s.
01:02
And to do that, we first differentiate s with respect to x.
01:08
And with that, we have s prime of x.
01:11
That's equal to 360x minus 40 x cubed.
01:17
And then from here, we want to set this to 0 and solve for x.
01:22
So we have 360x minus 40 x cubed equals 0.
01:27
We get 40 times x times 9 minus x squared equals 0 we get x equals 0 and then you have x squared equals 9 or that's x equals plus or minus 3 so the critical points are negative 3 0 and 3.
01:50
Next we want to do first derivative test so in the first derivative test we partition our number line using our critical points.
02:02
So we have this number line say from negative infinity to infinity and then we have critical points negative three zero and three.
02:12
Next we want to pick values in between the intervals formed by these boundaries.
02:20
So these are the intervals and then we will pick points in between.
02:25
So let's say you have negative four over here we can do negative 1, this one can pick 1, and for the last interval we can pick positive 4.
02:36
And then from here we want to find the sign of the first derivative...