Discuss the Continuity of the Function X4 Y4 X2 Y2
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discuss the continuity ofthe 2 functions below i. p (x, y) = cos (1/x) Sin (y) ii. g (x,y) = e(-x^2)(-y^2) find the maximum, minimum, and saddle points for the surface h (X, y) = 4xy-y4-x4 show all partial derivatives. (there are only 3 easy choices of critical values)
Related Question
Determine the critical points of the functions below and find out whether each point corresponds to a relative minimum, maximum, saddle point or no conclusion can be made. 1. f(x , y) = x2 + 3 y2 - 2xy - 8x 2. f(x , y) = x3 - 12 x + y3 + 3 y2 - 9y
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Video Transcript
Hello everyone. So this is the question that we have. We need to determine the critical points of the relative minimum maximum maximum saddle points or no conclusion. Now let's jump onto the solution of the question. So we have the equations asked part one, F X comma Y is equal to X squared plus three Y squared minus two X Y minus the attacks. Part two is F X. Of y is equal to X q minus 12 X plus y Q plus three Y squared minus nine. Y. No if we have these two equations but why? So first of all take ffx fx This is gonna be two weeks plus Why? There is no constraints or zero minus two Y minus eight. So this equals two, two words minus to wear. Similarly We'll take out FY. Which is going to be 6 1 -2 weeks. Then F X X. This is gonna be too FYY. Which is going to be six blast F X Y. Which is -2 Now FX zero. So I'm going to have the equation as say F X equal to zero and F Y is equal to zero. So the same place, I'll have the equation as six Y minus two X. Is equal to zero and two weeks minus two away -8 is equal to presume. So from solving these two equations I'm going to get X is equal to six And why is equal to two. So my coordinates XY are gonna become six. So my distance at 6:02 is there gonna be given by F X X. Half way way minus F. X. Whether. Why? So this is gonna be two multiplied by six minus negative two whole square. Which is equal to eight and eight is greater than zero. So which means six to hazard local minimum. It has a local minimum. And if it has a local minimum, so I'd say my function at this 0.6, homer two would be equal to six square plus three, multiplied by two square mile, multiplied by six, multiplied by two, multiplied by six. Which is going to be -24. I'm gonna have a local minimum, add .6 -24. This is my local minimum. Now coming onto the second equation that we have. Cool, this is gonna be the equation Second. We are going to use a similar technique. So fx is going to be three X square minus 12. F Y. Three Y squared plus six y minus nine. Then f X X. There's gonna be 66. F Y y is gonna be 65 plus six. FX Ray is going to be zero. But if X is equal to zero. So fx is equal to a is equal to zero. So this is going to be three. Y square Plus six way -9 is equal to zero and three X squared minus 12 is equal to zero. So I'm going to have the value of Y is equal to -301. So my coordinates X Y would become minus two. It goes from here. X would be either less or negative two. So minus two. I'm going to have four coordinates minus two minus three minus 21 man to Obama -3. And then you know what? So I'm gonna have all the different coordinates and I'll take out all the points for different points. Two, everything would be different. And now if I look at now at point minus two comma minus three. So this is going to be minus 12, multiplied by minus six. So this is gonna be obviously positive of 72 which is greater than zero. So it is maximum at this point. Black saloon at negative two, negative three. Now similarly at minus two I'm a -1. So it has local maximum at this point. Your maximum at this point -2 -1. This is gonna be what Let's see -12 multiplied bed 12 which is -144 which is obviously less than zero. So it'll have a saddle point. It'll have a saddle point this group of this. So this is the starting point. No, the third point add two comma minus three. So this is going to be 12 multiplied by negative 12 Which is again -144 which is less than zero. So we love again a saddle point that my negative and too common that two comma one, it would be 12 multiplied by 12 which is 144 Greater than zero. So you have a local minimum, so minimum at two. I hope you understood the explanation. I'll see you in the next one.
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