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Consider the function f(€,y) = wly on the region z2 + y? < 9. Find the absolute extrema of f on the region using the traditional optimization method. Inclu...

Question

Consider the function f(€,y) = wly on the region z2 + y? < 9. Find the absolute extrema of f on the region using the traditional optimization method. Include list of all of the possible candidates for extrema Find the absolute extreme of f on the region using the Method of Lagrange Multipliers Include a list of all the possible candidates for extrema: Compare your answers to those obtained in part (a)-

Consider the function f(€,y) = wly on the region z2 + y? < 9. Find the absolute extrema of f on the region using the traditional optimization method. Include list of all of the possible candidates for extrema Find the absolute extreme of f on the region using the Method of Lagrange Multipliers Include a list of all the possible candidates for extrema: Compare your answers to those obtained in part (a)-



Answers

Use Lagrange multipliers to find the indicated extrema, assuming that $x, y,$ and $z$ are positive. Minimize $f(x, y, z)=x^{2}+y^{2}+z^{2}$ Constraint: $x+y+z-9=0$

Okay, So in this problem, we want to find some extreme values for the function if equals. Two X squared plus four wide script plus one in the region are defined as X y such that x square plus four wide square lesser equals to one. So first we can see there. Um mhm the imperial off the region. Uh, that means X square plus for White Square is strictly less than one. So we compute the partial derivative on the set off this past majority beacause zero. Yeah, we have a critical point. X equals why it causes hero, which is inside this, uh, interior. So for this point, f equals to one. And now we can see that the boundary case that means X square plus for white squares equal to one. So we use a lot of ground. Um, so in fact, we don't have to use the ground multiply method in this case, because on the boundary X square plus four y squared equals to what that means. F equals toe one plus one, which is toe own boundary Children. Um, so for any point on the boundary, if it is always equal to two, so the minimum point, The minimum value office one. It happens at the origin and the maximum value off because that's what happens on the boundary or any point on the boundary. So this is the final result. Mhm. It can be difficult.


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