In summary: Okay, I'll just provide the summary for now.In summary, the problem is to find the maximum and minimum values of ##f(x, y) = 4x + y^2## subject to the constraint ##2x^2 + y^2 = 4##. There are a few different ways to approach this problem, such as solving for one variable in the constraint and then substituting it into the function, or using Lagrange multipliers. However, there appears to be some confusion with the solutions, as x = 4 and y = undefined are incorrect. It would be beneficial to sketch the ellipse and understand the domain before attempting to solve the problem.
#36
Lifeforbetter
48
1
Pi-is-3 said:
Also, if you ever need, ## - \sqrt{a^2+b^2} \leq acos(\theta)+bsin(\theta) \leq \sqrt{a^2+b^2}##
Do you mean the proof? You can prove it through Lagrange multipliers, or their is a trigonometrical proof too. For the Lagrange multipliers proof, your optimization function is ##acos(\theta)+bsin(\theta)## and restrain function is ##cos^2(\theta)+sin^2(\theta)=1##.