Optimizing multivariable functions
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Second partial derivative test example, part 2
- [Voiceover] In the last video, we were given a multi variable function, and asked to find and classify all of its critical points. So critical points, just means finding where the gradient is equal to zero, and we found four different points for that. I have them down here. They were zero zero, zero negative two, square root of three and one, and negative square root of three and one. So then the next step is to classify those. And that requires the second partial derivative test. So what I'm gonna go ahead and do, is copy down the partial derivatives. Since we already computed those. Copy, and then just kind of paste them down here. Where we can start to use them for the second partial derivatives. So I'm gonna clean things up a little bit, and we don't need this simplification of it. So we've got our partial derivatives. Now since we know we want to apply the second partial derivative test. We've got to first just compute all of the different second partial derivatives of our function. That's just kind of the first thing to do. So let's go ahead and do it. The second partial derivative of the function with respect to x twice in a row. Will take the partial derivative with respect to x, and then do it with respect to x again. So this first term looks like six times a variable times a constant, so it'll just be six times that constant. And then the second term. The derivative of negative six x, is just negative six. Moving right along. When we do the second partial derivative with respect to y twice in a row. We take the partial derivative with respect to y, and then do it again. So this x squared term looks like nothing. It looks like a constant as far as y is concerned so we ignore it. The derivative of negative three y squared is negative six times y, and then the derivative of negative six y is just negative six. And then we can't forget that last crucially important mixed partial derivative term. Which is the partial derivative of f. Where first we do it with respect to x, and then with respect to y. The order doesn't really matter in this case since it's a perfectly ordinary polynomial function. So we could do it either way, but I'm just gonna choose to take a look at this guy and differentiate it with respect to y. So the derivative of the first term with respect to y is six x, six x. And then that second term looks like a constant with respect to y, so that's all we have. So now what we're gonna do is plug in each of the critical points to the special second partial derivative test expression. And to remind you of what that is. That expression is we take the second partial derivative with respect to x twice. And I'll just write it with a kind of shorter notation using subscripts. Then we multiply that by the second partial derivative with respect to x, and then we subtract off. Subtract off the mixed partial derivative term squared. So let's go ahead and do that for each of our points. So when we do this at the point zero zero. Zero zero, what we end up getting. Plugging that into the partial derivative with respect to x twice. Six times zero is zero, so that's just negative six. So that gives us negative six multiplied by. When we plug it into this partial derivative with respect to y squared. Again, that y goes to zero. So we're left with just negative six. And then we subtract off the mixed partial derivative term. Which in this case is zero. Cause when we plug in x equals zero, we get zero. So we're subtracting off zero squared. And that entire thing equals negative six times negative six is 36, 36. And we'll get to analyzing what it means that that's positive in just a moment, but let's just kind of get all of them on the board so we can kind of start doing this with all of them. If we do this with zero and negative two. Zero and negative two. Then once we plug in y equals negative two to this expression. This time I'll write it out. Six times negative two minus six, so that's negative 12 minus six. We'll get negative 18, negative 18. Then when we plug it into the partial derivative of f with respect to y squared. Again, I'll kinda write it out. We have negative six times y is equal to negative two minus six. So now we have negative six times negative two, so that's positive 12 minus six. So that will be a positive six that we plug in here. And then for the mixed partial derivative. Again, x is equal to zero. So the mixed partial derivative is just gonna look like zero when we do this. So we're subtracting off zero squared and we get negative 18 times six. And geez what's 18 times six. So that's gonna be 36 times three. So that's the same as 90. Plus 18, so I think that's 108. Negative 108, and the specific magnitude won't matter. It's gonna be the sign that's important. And this is definitely negative. So now kind of moving right along. These examples can take quite a while. If we plug in square root of three one. Square root of three one, what we get. Now instead of plugging in y equals negative two. We're plugging in y equals one. So that'll be six times one minus six. So the whole thing is just zero. And then for the partial derivative with respect to y squared. Instead of plugging in negative two. Now we're plugging in y equals one. So we have negative six times one minus six so the whole thing is negative 12. So negative 12, and now for the mixed partial derivative term. Which is six x. X is equal to the square root of three. So now we're subtracting off the square root of three squared. So what that equals is, this first part is just entirely zero, and we're subtracting off three. So that's negative three. And then we have square root of three. No, no we don't, that's what we just did. Now we have negative square root of three one. And this will be very similar 'cause this first term just had a y and we plugged in a y. So it's also gonna be zero. For totally the same reasons, and same deal over here. The value of y didn't change. So that's also gonna be negative 12. Doesn't really matter cause we're multiplying it by zero, right? And then over here, now we're plugging in negative square root of three, and that's gonna have the same square. So again we're just subtracting off three. So what does the second partial derivative test tell us? Once we express this term. If it's greater than zero. We have a max or a min. That's what the test tells us. And then if it's less than zero. If it's less than zero we have a saddle point. So in this case, the only term that's greater than zero is this first one, is this first one. And to analyse whether it's a maximum or a minimum. Notice that the partial derivative with respect to x twice in a row, or with respect to y twice in a row was negative. Which indicates a sort of negative concavity. Meaning this corresponds to a maximum. So this guy corresponds to a local maximum. Now all of the other three gave us negative numbers. So all of these other three give us saddle points. Saddle points. So the answer to the question. The original find and classify such and such points is that we found four different critical points. Four different critical points. Zero zero, zero negative two, square root of three one, and negative square root of three one. And all of them are saddle points except for zero zero. Which is a local maximum. And all of that is something that we can tell without even looking at the graph of the function. And with that I will see you next video.