00:01
So in this problem, we create a little bubble around, or a little sphere, around the point 2 .1, 2, 1, 2 .1, 2.
00:15
And by that, we let g of x, comma, y, z be equal to x minus 2 squared plus y minus 1 squared plus z minus 2 squared this is a little sphere of radius 8 centered at 212 and f well we already are told that f x comma y comma z is um well we can make this equal to x squared plus y squared plus z squared and we know that this is equal to one and the gradient of capital left will be the vector 2x comma 2y comma 2 z the gradient of capital g is equal to 2 times x minus 2, comma 2 times y minus 1, comma 2 times z minus 2.
01:14
And let the gradient of g be equal to lambda times the gradient of f.
01:20
In this case, we are going to end up with 2 times x minus 2 equal to 2 lambda x.
01:27
2 times y minus 1 is equal to 2 times lambda y and 2 times z minus 2 is going to equal to 2 times lambda times z so what does it mean in this case well 2 can cancel out and we're going to end up with x minus 2 equals to lambda x that would mean that if you want to solve for x that x times 1 minus lambda lambda is going to equal to 2 which will mean that x is equal to 2 divided by 1 minus lambda similarly when we solve for y here the 2s can cancel out so we end up with y times 1 minus lambda to equal to y which will imply that that equals to 1 we correct myself there which implies y is equal to 1 over 1 minus lambda and for z is the same case as that that of for x because we're going to end up with z to equal to two divided one minus lambda so what do we do this pretty much we can try to figure it out what the value of lambda would be but plugging this into x squared plus y square plus z squared equals 1 which will mean that 4 over 1 minus lambda squared plus 1 over 1 minus lambda squared plus 1 over 1 minus lambda squared that this would equal to 1.
03:06
So that would mean that 4 plus 1 is 9th, that 1 minus lambda squared is equal to 9th, which means that 1 minus lambda is plus or minus 3...