In many applications we want to maximize or minimize some

Question

Question: In many applications, we want to maximize or minimize some quantity subject to a condition. Such constrained optimization problems are solved using Lagrange multipliers in multivariable calculus;

Minimize x² + y² while satisfying x + y = 4 using the following steps.

(a) Graph x + y = 4. On the same axes, graph x² + y² = 1, x² + y² = 4, x² + y² = 9.

(b) Explain why the minimum value of x² + y² on x + y = 4 occurs at the point at which a graph of x² +y² = Constant is tangent to the line x + y = 4.

(c) Using your answer to part (b) and implicit differentiation to find the slope of the circle, find the minimum value of x² + y² such that x + y = 4.