Solving optimization problems
Optimization problem: extreme normaline to y=x²
I just got sent this problem, and it's a pretty meaty problem. A lot harder than what you'd normally find in most textbooks. So I thought it would help us all to work it out. And it's one of those problems that when you first read it, your eyes kind of glaze over, but when you understand what they're talking about, it's reasonably interesting. So they say, the curve in the figure above is the parabola y is equal to x squared. So this curve right there is y is equal to x squared. Let us define a normal line as a line whose first quadrant intersection with the parabola is perpendicular to the parabola. So this is the first quadrant, right here. And they're saying that a normal line is something, when the first quadrant intersection with the parabola is normal to the parabola. So if I were to draw a tangent line right there, this line is normal to that tangent line. That's all that's saying. So this is a normal line, right there. Normal line. Fair enough. 5 normal lines are shown in the figure. 1, 2, 3, 4, 5. Good enough. And these all look perpendicular, or normal to the parabola in the first quadrant intersection, so that makes sense. For a while, the x-coordinate of the second quadrant intersection of the normal line of the parabola gets smaller, as the x-coordinate of the first quadrant intersection gets smaller. So let's see what happens as the x-quadrant of the first intersection gets smaller. So this is where I left off in that dense text. So if I start at this point right here, my x-coordinate right there would look something like this. Let me go down. my x-coordinate is right around there. And then as I move to a smaller x-coordinate to, say, this one right here, what happened to the normal line? Or even more important, what happened to the intersection of the normal line in the second quadrant? This is the second quadrant, right here. So when I had a larger x-value here, my normal line intersected here, in the second quadrant. Then when I brought my x-value in, when I lowered my x-value, my x-value here, because this is the next point, right here, my x-value at the intersection here, went-- actually, their wording is bad. They're saying that the second quadrant intersection gets smaller. But actually, it's not really getting smaller. It's getting less negative. I guess smaller could be just absolute value or magnitude, but it's just getting less negative. It's moving there, but it's actually becoming a larger number, right? It's becoming less negative, but a larger number. But if we think in absolute value, I guess it's getting smaller, right? As we went from that point to that point, as we moved the x in for the intersection of the first quadrant, the second quadrant intersection also moved in a bit, from that line to that line. Fair enough. But eventually, a normal line second quadrant intersection gets as small as it can get. So if we keep lowering our x-value in the first quadrant, so we keep on pulling in the first quadrant, as we get to this point. And then this point intersects the second quadrant, right there. And then, if you go even smaller x-values in the first quadrant then your normal line starts intersecting in the second quadrant, further and further negative numbers. So you can kind of view this as the highest value, or the smallest absolute value, at which the normal line can intersect in the second quadrant Let me make that clear. Up here, you were intersecting when you had a large x in the first quadrant, you had a large negative x in the second quadrant intersection. And then as you lowered your x-value, here, you had a smaller negative value. Up until you got to this point, right here, you got this, which you can view as the smallest negative value could get, and then when you pulled in your x even more, these normal lines started to push out again, out in the second quadrant. That's, I think, what they're talking about. The extreme normal line is shown as a thick line in the figure. Right. This is the extreme normal line, right there. So this is the extreme one, that deep, bold one. Extreme normal line. After this point, when you pull in your x-values even more, the intersection in your second quadrant starts to push out some. And you can think of the extreme case, if you draw the normal line down here, your intersection with the second quadrant is going to be way out here someplace, although it seems like it's kind of asymptoting a little bit. But I don't know. Let's read the rest of the problem. Once the normal line passes the extreme normal line, the x-coordinates of their second quadrant intersections what the parabola start to increase. And they're really, when they say they start to increase, they're actually just becoming more negative. That wording is bad. I should change this to more, more negative. Or they're becoming larger negative numbers. Because once you get below this, then all of a sudden the x-intersections start to push out more in the second quadrant. Fair enough. The figures show 2 pairs of normal lines. Fair enough. The 2 normal lines of a pair have the same second quadrant intersection with the parabola, but 1 is above the extreme normal line, in the first quadrant, the other is below it. Right, fair enough. For example, this guy right here, this is when we had a large x-value. He intersects with the second quadrant there. Then if you lower and lower the x-value, if you lower it enough, you pass the extreme normal line, and then you get to this point, and then this point, he intersects, or actually, you go to this point. So if you pull in your x-value enough, you once again intersect at that same point in the second quadrant. So hopefully I'm making some sense to you, as I try to make some sense of this problem. OK. Now what do they want to know? And I think I only have time for the first part of this. Maybe I'll do the second part in the another video. Find the equation of the extreme normal line. Well, that seems very daunting at first, but I think our toolkit of derivatives, and what we know about equations of a line, should be able to get us there. So what's the slope of the tangent line at any point on this curve? Well, we just take the derivative of y equals x squared, and y prime is just equal to 2x. This is the slope of the tangent at any point x. So if I want to know the slope of the tangent at x0, at some particular x, I would just say, well, let me just say, slope, it would be 2 x0. Or let me just say, f of x0 is equal to 2 x0. This is the slope at any particular x0 of the tangent line. Now, the normal line slope is perpendicular to this. So the perpendicular line, and I won't review it here, but the perpendicular line has a negative inverse slope. So the slope of normal line at x0 will be the negative inverse of this, because this is the slope of the tangent line x0. So it'll be equal to minus 1 over 2 x0. Fair enough. Now, what is the equation of the normal line at x0 let's say that this is my x0 in question. What is the equation of the normal line there? Well, we can just use the point-slope form of our equation. So this point right here will be on the normal line. And that's the point x0 squared. Because this the graph of y equals x0, x squared. So this normal line will also have this point. So we could say that the equation of the normal line, let me write it down, would be equal to, this is just a point-slope definition of a line. You say, y minus the y-point, which is just x0 squared, that's that right there, is equal to the slope of the normal line minus 1 over 2 x0 times x minus the x-point that we're at. Minus x, minus x0. This is the equation of the normal line. So let's see. And what we care about is when x0 is greater than 0, right? We care about the normal line when we're in the first quadrant, we're in all of these values right there. So that's my equation of the normal line. And let's solve it explicitly in terms of x. So y is a function of x. Well, if I add x0 squared to both sides, I get y is equal to, actually, let me multiply this guy out. I get minus 1/2 x0 times x, and then I have plus, plus, because I have a minus times a minus, plus 1/2. The x0 and the over the x0, they cancel out. And then I have to add this x0 to both sides. So all I did so far, this is just this part right there. That's this right there. And then I have to add this to both sides of the equation, so then I have plus x0 squared. So this is the equation of the normal line, in mx plus b form. This is its slope, this is the m, and then this is its y-intercept right here. That's kind of the b. Now, what do we care about? We care about where this thing intersects. We care about where it intersects the parabola. And the parabola, that's pretty straightforward, that's just y is equal to x squared. So to figure out where they intersect, we just have to set the 2 y's to be equal to each other. So they intersect, the x-values where they intersect, x squared, this y would have to be equal to that y. Or we could just substitute this in for that y. So you get x squared is equal to minus 1 over 2 x0 times x, plus 1/2 plus x0 squared. Fair enough. And let's put this in a quadratic equation, or try to solve this, so we can apply the quadratic equation. So let's put all of this stuff on the lefthand side. So you get x squared plus 1 over 2 x0 times x minus all of this, 1/2 plus x0 squared is equal to 0. All I did is, I took all of this stuff and I put it on the lefthand side of the equation. Now, this is just a standard quadratic equation, so we can figure out now where the x-values that satisfy this quadratic equation will tell us where our normal line and our parabola intersect. So let's just apply the quadratic equation here. So the potential x-values, where they intersect, x is equal to minus b, I'm just applying the quadratic equation. So minus b is minus 1 over 2 x0, plus or minus the square root of b squared. So that's that squared. So it's one over four x0 squared minus 4ac. So minus 4 times 1 times this minus thing. So I'm going to have a minus times a minus is a plus, so it's just 4 times this, because there was one there. So plus 4 times this, right here. 4 times this is just 2 plus 4 x0 squared. All I did is, this is 4ac right here. Well, minus 4ac. The minus and the minus canceled out, so you got a plus. There's a 1. So 4 times c is just 2 plus 4x squared. I just multiply this by two, and of course all of this should be over 2 times a. a is just 2 there. So let's see if I can simplify this. Remember what we're doing. We're just figuring out where the normal line and the parabola intersect. Now, what do we get here. This looks like a little hairy beast here. Let me see if I can simplify this a little bit. So let us factor out-- let me rewrite this. I can just divide everything by 1/2, so this is minus 1 over 4 x0, I just divided this by 2, plus or minus 1/2, that's just this 1/2 right there, times the square root, let me see what I can simplify out of here. So if I factor out a 4 over x0 squared, then what does my expression become? This term right here will become an x to the fourth, x0 to the fourth, plus, now, what does this term become? This term becomes a 1/2 x0 squared. And just to verify this, multiply 4 times 1/2, you get 2, and then the x0 squares cancel out. So write this term times that, will equal 2, and then you have plus-- now we factored a four out of this and the x0 squared, so plus 1/16. Let me scroll over a little bit. And you can verify that this works out. If you were to multiply this out, you should get this business right here. I see the home stretch here, because this should actually factor out quite neatly. So what does this equal? So the intersection of our normal line and our parabola is equal to this. Minus 1 over 4 x0 plus or minus 1/2 times the square root of this business. And the square root, this thing right here is 4 over x0 squared. Now what's this? This is actually, lucky for us, a perfect square. And I won't go into details, because then the video will get too long, but I think you can recognize that this is x0 squared, plus 1/4. If you don't believe me, square this thing right here. You'll get this expression right there. And luckily enough, this is a perfect square, so we can actually take the square root of it. And so we get, the point at which they intersect, our normal line and our parabola, and this is quite a hairy problem. The points where they intersect is minus 1 over 4 x0, plus or minus 1/2 times the square root of this. The square root of this is the square root of this, which is just 2 over x0 times the square root of this, which is x0 squared plus 1/4. And if I were to rewrite all of this, I'd get minus 1 over 4 x0 plus, let's see, this 1/2 and this 2 cancel out, right? So these cancel out. So plus or minus, now I just have a one over x0 times x0 squared. So I have 1 over x0-- oh sorry, let me, we have to be very careful there-- x0 squared divided by x0 is just x0, let me do that in a yellow color so you know what I'm dealing with. This term multiplied by this term is just x0, and then you have a plus 1/4 x0. And this is all a parentheses here. So these are the two points at which the normal curve and our parabola intersect. Let me just be very clear. Those 2 points are, for if this is my x0 that we're dealing with, right there. It's this point and this point. And we have a plus or minus here, so this is going to be the plus version, and this is going to be the minus version. In fact, the plus version should simplify into x0. Let's see if that's the case. Let's see if the plus version actually simplifies to x0. So these are our two points. If I take the plus version, that should be our first quadrant intersection. So x is equal to minus 1/4 x0 plus x0 plus 1/4 x0. And, good enough, it does actually cancel out. That cancels out. So x0 is one of the points of intersection, which makes complete sense. Because that's how we even defined the problem. But, so this is the first quadrant intersection. So that's the first quadrant intersection. The second quadrant intersection will be where we take the minus sign right there. So x, I'll just call it in the second quadrant intersection, it'd be equal to minus 1/4 x0 minus this stuff over here, minus the stuff there. So minus x0 minus 1 over 4 mine x0. Now what do we have? So let's see. We have a minus 1 over 4 x0, minus 1 over 4 x0. So this is equal to minus x0, minus x0, minus 1 over 2 x0. So if I take minus 1/4 minus 1/4, I get minus 1/2. And so my second quadrant intersection, all this work I did got me this result. My second quadrant intersection, I hope I don't run out of space. My second quadrant intersection, of the normal line and the parabola, is minus x0 minus 1 over 2 x0. Now this by itself is a pretty neat result we just got, but we're unfortunately not done with the problem. Because the problem wants us to find that point, the maximum point of intersection. They call this the extreme normal line. The extreme normal line is when our second quadrant intersection essentially achieves a maximum point. I know they call it the smallest point, but it's the smallest negative value, so it's really a maximum point. So how do we figure out that maximum point? Well, we have our second quadrant intersection as a function of our first quadrant x. I could rewrite this as, my second quadrant intersection as a function of x0 is equal to minus x minus 1 over 2 x0. So this is going to reach a minimum or a maximum point when its derivative is equal to 0. This is a very unconventional notation, and that's probably the hardest thing about this problem. But let's take this derivative with respect to x0. So my second quadrant intersection, the derivative of that with respect to x0, is equal to, this is pretty straightforward. It's equal to minus 1, and then I have a minus 1/2 times, this is the same thing as x to the minus 1. So it's minus 1 times x0 to the minus 2, right? I could have rewritten this as minus 1/2 times x0 to the minus 1. So you just put its exponent out front and decrement it by 1. And so this is the derivative with respect to my first quadrant intersection. So let me simplify this. So x, my second quadrant intersection, the derivative of it with respect to my first quadrant intersection, is equal to minus 1, the minus 1/2 and the minus 1 become a positive when you multiply them, and so plus 1/2 over x0 squared. Now, this'll reach a maximum or minimum when it equals 0. So let's set that equal to 0, and then solve this problem right there. Well, we add one to both sides. We get 1 over 2 x0 squared is equal to 1, or you could just say that that means that 2 x0 squared must be equal to 1, if we just invert both sides of this equation. Or we could say that x0 squared is equal to 1/2, or if we take the square roots of both sides of that equation, we get x0 is equal to 1 over the square root of 2. So we're really, really, really close now. We've just figured out the x0 value that gives us our extreme normal line. This value right here. Let me do it in a nice deeper color. This value right here, that gives us the extreme normal line, that over there is x0 is equal to 1 over the square root of 2. Now, they want us to figure out the equation of the extreme normal line. Well, the equation of the extreme normal line we already figured out right here. It's this. The equation of the normal line is that thing, right there. So if we want the equation of the normal line at this extreme point, right here, the one that creates the extreme normal line, I just substitute 1 over the square root of 2 in for x0. So what do I get? I get, and this is the home stretch, and this is quite a beast of a problem. y minus x0 squared. x0 squared is 1/2, right? 1 over the square root of 2 squared is 1/2. Is equal to minus 1 over 2 x0. So let's be careful here. So minus 1/2 times 1 over x0. One over x0 is the square root of 2, right? All of that times x minus x0. So that's 1 over the square root of 2. x0 is one over square root of 2. So let's simplify this a little bit. So the equation of our normal line, assuming I haven't made any careless mistakes, is equal to, so y minus 1/2 is equal to, let's see. If we multiply this minus square root of 2 over 2x, and then if I multiply these square root of 2 over this, it becomes one. And then I have a minus and a minus, so that I have a plus 1/2. I think that's right. Yeah, plus 1/2, this times this times that is equal to plus 1/2. And then, we're at the home stretch. So we just add 1/2 to both sides of this equation, and we get our extreme normal line equation, which is y is equal to minus square root of 2 over 2x. If you add 1/2 to both sides of this equation, you get plus 1. And there you go. That's the equation of that line there, assuming I haven't made any careless mistakes. But even if I have, I think you get the idea of hopefully how to do this problem, which is quite a beastly one.