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Virendrasingh D.

If $f(x)=x+\sqrt{2-x}$ and $g(u)=u+\sqrt{2-u},$ is it true that $f=g ?$

01:59

Nutan C.

00:38

Amy J.

05:28

Felicia S.

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All right, so now let's have a little bit of fun. So we want to find the point on this fear that is closest and furthest away from this point. One comma, one comment to So the function that we're maximizing and minimizing. Or, in other words, our objective function is going to be the distance from this point right here. And it's fine if we just go ahead and actually use the distance squared to get rid of the square root. The points that we find will be the same, so the distance squared is going to be X minus. One squared. Why minus one squared plus C minus two squared in our constraint is X squared plus y squared plus Z squared. And we need that constraint function to be equal to one. Okay, so let's set the Grady INTs equal to each other or sorry, equal to a multiple of each other. So the grading of death we want that to be a multiple scaler multiple of the Grady int of G. And so what do we get? Well, the grading of F is two times X minus one two times why minus one and then two times Z minus two And then we have Lambda in the Grady in of G is two x to Why to z All right, so we get three equations on this, and our fourth equation is going to be that the constraint needs to be one. I noticed that we actually have four unknown variables, so our our system should be solvable. So the equations we get, I can divide by two Computing the x coordinate. So I have X minus one equals Lambda X. I have y minus one equals lambda. Why? And then I have Z minus two equals Lambda Z. So if I take these first two equations and subtract them what I get so X minus one minus Y minus one is X minus y And on the right hand side, Lambda X minus lambda. Why is Lambda Times X minus y? But this is excellent because I see that either one of two things can happen either. Well, X could equal why which makes both sides of this equation zero. So that was But that would be true, or X and y could be anything, and lambda could be one. So either x x equals y or land A is equal toe one. But what happens if Lambda equals one? Let's look at the first equation. We have that X minus one will equal X. But then, if I subtract eggs, I get minus one equals zero. But this is a contradiction. So that means Lambda equals one can't happen. So then it must be the case that X is equal toe. Why, Okay, so now that we know the exes gold or why, let's try to figure out something about Z. So if I saw this first equation for Lambda, I get lambda is equal to X minus one over X and let me plug that in for Atlanta in the Z equation. So I have Z minus two is equal to X minus one over X times Z or if I multiply both sides by X I am Z X minus two X equals Z X minus c The Z X is I can cancel and I get that Z is equal to two X. Well, that's great. Now I know that X equals Y and Z is equal to two X. So now let me plug all that back in into my constraint function. So I have X squared, plus why it's just X. So another X squared and then Z is two X, so that's going to be four x squared when I square Z is equal to one. Or, in other words, six X squared equals one or X is equal to plus or minus square root of 1/6 and notice that that's also what? Why, yes, because X is equal toe. Why? So I get that? Why is equal to plus or minus square root of 1/6, and then C is twice X. Susie is plus or minus to times the square root of 1/6. All right, And so notice that if X is plus squared of +16 why will be plus squared of +16 and Z is plus two times the square of +16 and so the closest point to this 0.112 is going to be the positive solution, so the closest is going to be square root of 16 square of 16 to square root of 16 That's because that's the positive point. That's the point in the positive first oxidant, which is going to be in the same opt in at this point, and then the furthest point is going to be actually the negative solution. So that's going to be minus square root. 16 This is going to be on the other side or the opposite side of the sphere. Nice to square 16 So here's my closest point. Here's my furthest point.

Multiple Integrals

Vector Calculus

04:59

08:13