Question

Given 1 is less than or equal x^2 +y^2 less than or equal 2, find the largest and smallest values of the expression z=x^2 +xy+y^2

Answer 1

I like to think of a circle in the xy plane where the radius is between 1 and the square root of two.

Answer 2

(centered at zero)

Answer 3

The interesting places are where x=y and x = -y

Answer 4

this is like Lagrange multiplier twice

Answer 5

but the problem is its an inequality