Why function is maximum at intersection of the level curve with its constrained curve?

Question

What I'm asking is the core question of Lagrange multiplier. Say we have a function $f:\mathbb{R^2}\rightarrow \mathbb{R}$. Now we intend to calculate at what $(x,y)$, the function achieves its maxima given constraint $g:\mathbb{R^2}\rightarrow\mathbb{R}$. The Lagrange multiplier says that we will find the maxima of the function where the level curve intersects the constrained function at exactly one point. Why is that? I'm not able to understand that why the function will achieve its maxima at that particular point

Answer 1

The following is a standard explanation.

Consider this specific example: $$\max f(x, y) = xy$$ $$\text{s.t. } x^2 + y^2 = 1$$

Let $L_c = \{(x, y)|f(x, y) = c\}$ be the level curve of $f$ at $c$. This is how the picture looks for $c = 4.3$:

As we decrease the value of $c$, the hyperbola comes closer and closer to the origin (you can interact with the graph here.) If we shrink the hyperbola until it is tangent to the circle, we will have a feasible point whose $f$ value is pretty high. But if we decrease $c$ even more, there will be $4$ points of intersection, but the objective value will be unnecessarily low.