Prove that $C=\{(x,y) \in \mathbb{R^2} : 4xy - 3y^2 = 20 \}$ is $ C^\infty$ smooth in a neighbourhood of the point $(4,2)$

Question

Prove that $C=\{(x,y) \in \mathbb{R^2} : 4xy - 3y^2 = 20 \}$ is $ C^\infty$ smooth in a neighbourhood of the point $(4,2)$

Any hints on how to prove this? I'm somewhat lost, the only thing I can think of would be to use something related to Taylor series

Answer 1

Let $f:\mathbb{R}^2\rightarrow \mathbb{R}$ be the function $$f(x,y) = 4xy-3y^2$$

To show that the level set $f(x,y)=20$ is smooth in a neighborhood of the point $(4,2)$ it suffices to show that $Df_{(4,2)}$ has rank 1. This is because $f$ is smooth, and if the derivative has rank 1 at the point, we can apply the implicit function theorem. So first, we compute $Df_{(4,2)}$

$$Df_{(x,y)}= [ 4y, 4x-6y]$$ $$Df_{(4,2)}= [ 8, 4]$$

Clearly, $Df$ has rank 1 at $(4,2)$. Hence, the implicit function theorem say the level set is smooth in a neighborhood of $(4,2)$.