单个约束下的拉格朗日乘数法

The champion on the constraint has parallel gradients

A worked constrained problem

The multiplier carries information

Practical notes on solving the system

To optimize $f(x, y) = x + 2y$ subject to $x^2 + y^2 = 5$, which Lagrange system is correct?

Solving the system gives candidates $(1, 2)$ and $(-1, -2)$ for $f = x + 2y$ on $x^2 + y^2 = 5$. What are the constrained maximum and minimum?

At a constrained optimum, the multiplier is found to be $\lambda = 0$. What does this report?

Among points of the circle $x^2 + y^2 = 5$, the point $(1, 2)$ maximizes $f = x + 2y$. Give the value of $f$ there, as a plain number.

A constrained optimum has $\lambda = 3$, and the constraint level is nudged from $c = 10$ to $c = 10.4$. Estimate the change in the optimal value of $f$, as a decimal like 1.2.