A subtle problem regarding tangent line when gradient is zero

Question

In the section of "Applications of Partial Derivates" in Adams' calculus book, 7th edition, page 756, there is a part which talks about "Lagrange Multipliers". It says:

Suppose that $f$ and $g$ have continuous first partial derivatives near the point $P_0=(x_0,y_0)$ on the curve $C$ with equation $g(x,y)=0$. Suppose also that, when restricted to points on $C$, the function $f(x,y)$ has a local maximum or minimum value at $P0$. Finally suppose that:
(i) $P_0$ is not an endpoint of $C$, and
(ii) $∇g(P_0)≠0$,
Then there exists a number $λ_0$ such that $(x_0,y_0,λ_0)$ is a critical point of the Lagrangian function $L(x,y,λ)=f(x,y)+λg(x,y)$.

Proof: Together (i) and (ii) imply that $C$ is smooth enough to have a tangent line at $P_0$…

Can someone explain why the first sentence of the proof is true? I mean why is it that (i) and (ii) imply that $C$ is smooth enough to have a tangent line? Note that my main problem is with (ii). What happens to the smoothness and tangent line when $∇g(P_0)$ is zero and what difference does it make when it is non-zero? Is it a necessary condition for smoothness of $C$ on $P_0$ that $∇g(P_0)≠0$?

Any help on the relevance of (ii) to the smoothness of $C$ on $P_0$ would be highly appreciated.

Answer 1

$\def\R{\mathbb{R}}\def\d{\mathrm{d}}$For your first question, because$$ g(x, y) = 0 \Longleftrightarrow \frac{∂g}{∂x}(x, y) \,\d x + \frac{∂g}{∂y}(x, y) \,\d y = 0, $$ if $∇g(x_0, y_0) ≠ (0, 0)$, then the tangent line of $C$ at $P_0$ is$$ \frac{∂g}{∂x}(x_0, y_0) (x - x_0) + \frac{∂g}{∂y}(x_0, y_0) (y - y_0) = 0. $$

For your second question, it is indeed not necessary to require $∇g(x_0, y_0) ≠ (0, 0)$ to ensure that $C$ have a tangent line at $(x_0, y_0)$. For example, if $g(x, y) = (x^2 - y)^2$ for $(x, y) \in \R^2$ and $P_0 = (0, 0)$, then $C = \{(t, t^2) \mid t \in \R\}$ has the tangent line $y = 0$ at $P_0$, but $∇g(0, 0) = (0, 0)$.

However, the condition $∇g(x_0, y_0) ≠ (0, 0)$ is usually imposed to ensure that $C$ does not bifurcate at $P_0$. When this condition is statisfied, at least one of the following proposition holds by the implicit function theorem:

1. There exists a function $u$ such that $g(x, y) = 0 \Leftrightarrow y = u(x)$ for $(x, y)$ in a neighborhood of $P_0$;
2. There exists a function $v$ such that $g(x, y) = 0 \Leftrightarrow x = v(y)$ for $(x, y)$ in a neighborhood of $P_0$,

which implies that $P_0$ is not a self-intersection point of $C$. For example, if$$ g(x, y) = (x^2 + y^2)^2 - (x^2 - y^2),\quad \forall (x, y) \in \R^2 $$ and $P_0 = (0, 0)$, then $∇g(0, 0) = (0, 0)$ and $C$ self-intersects at $P_0$.