minimizing and maximizing function with constraint

Question

I'm studying for my calc II exam, and I recently discovered how one can solve max/min problems with constrains using AM-GM inequalities instead of lagrange multipliers. Seeing as this is way more elegant, I'd like to learn more about it.

Could someone help me find the maximum and minimum value of $3+2xy$ subject to the constraint $x^2+y^2=1$. I've already solved it using lagrange multipliers, but I'd like to see how it can be solved using AM-GM aswell.

(I've only looked at the very basics of AM-GM so far, but I couldn't find much online about how to utilize the inequalities when you have a constraint. Is this worth learning before my exam?)

Thank you!

Answer 1

From $$x^2+y^2=1$$ we get $$(x+y)^2=1+2xy\geq 0$$ For the Maximum prove that $$3+2xy\le 4$$

Answer 2

Alternative, polar coordinates.

$x=r \cos t$, $y=r\sin t$, $0\le t \lt 2π.$

Note: $r =1.$

$F(r,t):= 3+2\sin t \cos t =$

$3+ \sin 2t.$

Minimum, maximum?