We have a number 35. The function given is X over one place access choir plus y squared. So first we need to get the critical points for that. We need to differentiate this function partially with respect to X and equated equal to zero. So this is in the form off you buy we so it's partial differentiation will be one plus X squared plus y squared into differentiation Off one is one Sorry, access one minus. We will be keeping X as it is and differentiation off this quantity with respect toe ex keeping wise constant. So two weeks he went by one plus x squared plus y square in tow, Whole square, equal to zero. So, uh, if you cross multiply and opening the back it this will be one plus x squared plus y squared minus two access quite equal to zero, which means one minus X squared plus y squared equal to zero. This is a very question number one. Similarly, we have to differentiate the function partially with respect to y, and it created equal toe zero. So, no, this will be one plus x square plus y square in +20 because we have X in the numerator and with inspector y axis constant. So this will be zero differentiation of X with respect to y zero minus x will demand as it is one plus. I'm sorry how we have to differentiate this So one will have zero differentiation. Placido plus two y divided by one place Access square plus y square, Whole square, equal to zero. Okay, so now we will be having two x y called to zero so X equal to zero. And why I called to zero. Okay. And we have a question. Number 11 minus access squirrel plus y squared equal to zero. So let us plug in. Let us Ah, look at this. When x equal to zero, this will become one minus zero plus y squared equal to zero. So one plus y squared equal to zero. Not possible. Okay. And when why call to zero? We should plug in here. So one of minors access choir plus zero equal to zero. So x will be called toe plus minus one. So we'll be having two critical points when X is minus one y zero when X is one wise zero. So these are the critical points. Okay, No. We will do the world differences in, or we can say partially transition off FX with respect to X. Our double partial differentiation off FX four with respect to X. So we have to differentiate this quantity. Okay. And this content is one minus x squared plus y squared by mhm. This is one minus X squared plus y squared by one plus x squared. That's why a square hole square. We have to differentiate with respect to X. So this will be one plus x square plus wire square. This will be taken as it is and differentiation off the numerator with respect works. So minus two x minus, we will write it as it is and differentiation off the denominator. So this will be two in tow. One plus access choir plus y squared in 22 x, he right by one plus access choir plus y squared whole square. Okay. No, we'll be plugging in the values. Okay. Now we'll be finding value off this expression at minus one. Comma zito so x equal to minus one. This will become one plus 12 to square into tau minus one minus 100 So this whole thing will become zero. Okay, so we have only this divide by the C one plus 12 square that is to square. So this will get canceled out. So when you is to similarly uh huh. This is too similarly value off the same quantity. Same expression with respect at one common zero. So well, we will be one plus 1 to 2 square in two minus two, and this will become zero. Divide by two square. So this is minus two couple. Similarly, we have to find double differentiation. Possible partial differentiation. With respect to why so simply we have to differentiate this expression with respect away. So after, uh, solving this will be getting minus two x y by one place successful plus wise under minus two x y. We have to do Carly bike Oliva minus two x y by one place access choir plus y squared whole square. Okay, so minus two will get canceled minus two will get outside. And our expression should be one place. That's a squared. Plus wise hold square in two. We're differentiating with respect to wise, actually constant. So we'll be one. Sorry, will be excellently minus X Y into two. And two, uh, 00 So do why. Okay, something is fishy. One plus x squared, plus y squared will be here. One plus access choir. Plus why square then too. I will be okay. Do I buy one place? Access Square plus Y Squire raised the powerful. Okay. Okay. Here. It was some mistake. It should have been raised about four, so it should have been raised to the power. This was all right to risk wire by tourists to the power four. So it will be won by two. Similarly Yeah, it will be to rest in part four. So this will be minus one by four. I won by two. Sorry. Minus one by toe. Right. Okay. No. Well, you have this expression at our critical points, which is have why we're at minus one, comma zero. So this will be minus two one plus 12 minus two. Okay, this is minus two into, uh, zero. That's a minus two. This whole thing will be zero. So we'll be having one plus one to minus two minus two. Divide by one and 1 to 2 weeks to the par four. So we'll be having fall by 16. That is one by four. Okay, this is one before similarly, the value of this expression at our critical point. Why? Why one comma zero we have to plug in. So minus two will be as it is. This will be one plus one toe to Squire. Okay. Okay. Here. You should have been toe Squire, because this is quiet to a square. So four minus eight. So minus one by three, it should have been minus one by three. Because this is to to score four with eight so minus one by three. Similarly, here it will be too. And two is quiet into one. So similarly, it will be to square. The whole thing will be zero and divide by two days to the power four. So the saloon minus one by three. And here it will be extras minus one. Okay, So negative will be outside. Negative will be outside. So will be positive. Yes, we have got these values. Now we have to calculate one more thing, which is mhm but partial differentiation off this effects will with respect to why so we have to differentiate partially this quantity with respect to y. Okay, this quantity that is one minus X squared plus y squared, divided by one plus x squared plus y squared hold square or we can divide We can differentiate this quantity toe x y by one plus x squared plus y squared old square. So let us differentiate 1st. 1st only first month. This that is one of my insects. Square place where Squire by one plus x squared. Plus y Okay, one minus. Access choir plus fire squared, divided by one plus access choir plus y squared whole square with respect to y. So this will be one place access choir plus y squared one plus x squared plus y square. And this was Zito zero to why minus one minus X squared plus y square. We have, uh, kept this as it is indifference. We lifted and we will be differentiating this with respect to y so. To one plus x squared plus y square in two. Do what? Okay, this was whole square. Okay, Do I buy one place? Access choir plus y squared whole square. So value of this expression at minus one comma zero. That is first critical point will be C minus the this will be zero all become zero and we'll be having negative off one of minus 100 So everything will be zero zero and value off this the same expression with respect to why at one comma, zero will be This is the 1010 So this will also be zero. So both values are zero. Now we have we know that discriminate and at X come away in f xx f. Why? What minus? Fx, Where Whole square. So at critical point minus one comma zero Let us find the discriminate. The minus one comma zero D minus one comma zero is F XX at minus one comma zero. So this is one by two for F Y. Y is one by three. One by two into one by three minus zero That is one by six and we have f x x at miles one comma zero equal to one by two. So this is also greater than zero. This is also greater than zero. So both greater than zero which means the function have local minima local minima at, uh this critical point that is minus one comma zero. Now, if you talk about critical 0.1 comma zero one comma zero, that is second critical point D minus one Comma zero will be equal. Toe minus one by two. Minus one by three minus zero. That is one by six and F x x at minus one. Comma zero. Oh, sorry. One comma zero. Okay, it waas one comma zero minus one by two. This should be on commodity, not medicine. Comma zero minus one by two. So this is positive. This is negative. So local Maxima at one comma. Zero. Thank you so much.