10.27.3 Can you answer these questions?

10.27.3.1 `Needs` versus `<<`

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=306169

Is it possible that loading an .m package via <<ReactionKinetics  produces errors the second time it is loaded, while Needs["ReactionKinetics "] doesn'... 
[packages]
 

asked by florin https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=54979 Score of 1

10.27.3.2 FilledPolarCurve: Area and Integrate give different values

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=306227

Version 14.1 introduces FilledPolarCurve as a new way to make regions. 
In some cases, Area and integrating 1 over the region give different values for the region's area. Is this a bug? 
Example: 
... 
[calculus-and-analysis] [bugs] [regions] [area]
 

asked by tad https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=70428 Score of 3

10.27.3.3 How to use Lagrange multipliers to find the maximum value with constraints?

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=306329

How can the Lagrange multipliers method be employed to optimize the function?$f(x_1,x_2)=(x_1^p+x_1^p)^{1/p}$ 
subject to the constraint $g(x_1,x_2)=p_1x_1+p_2 x_2-y=0$ 
I am a beginner, and the results ... 
[equation-solving] [mathematical-optimization]
 

asked by lemmingxuan https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=101152 Score of 1

answered by Daniel Huber https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=46318 Score of 0