Consider the real-valued function $f(x, y):= x + y$ defined on points $(x, y)$ in a $2-$dimensional Euclidean plane.
Suppose you want to find all possible $(x^{\ast}, y^{\ast})$ that maximize the value of the function $f\left(\cdot,\cdot\right)$ subject to the condition that $(x^{\ast})^2 + (y^{\ast})^2 = 2.$
State all possible solutions $\left(x^{\ast}, y^{\ast}\right):$ ______
Suppose you want to find all possible $(x', y')$ that maximize the value of the function $f(\cdot,\cdot)$ subject to the condition that $|x'|^{0.9} +|y'|^{0.9} = 1.$
State the value of the function $f(\cdot,\cdot)$ at all possible solutions $(x', y'):$______
