Intermediate value Theorem for Curve in $\mathbb{R}^2$

Question

I am stuck on this problem from a previous exam.

Let $f:\mathbb{R}^2 \to \mathbb{R}$ a $C^1$ function.

Assume:

1. $C=\{f(x,y)=0\}\subset \mathbb{R}^2$ is a path-connected curve.
2. $\forall (x,y) \in C : f'|_{(x,y)}\neq 0$.
3. $C$ is tangent to the $\hat{x}$-axis at $(1,0)$, and to the $\hat{y}$-axis at $(0,1)$.

Prove:

C is tangent to a line $x+y=d$ for some $d\in \mathbb{R}$.

Using the Implicit Function Theorem I have derived that $gradf|_{(1,0)} = (0,\alpha)$, and $gradf|_{(0,1)} = (\beta,0)$ for some $\alpha, \beta \in \mathbb{R}$.

I would like to use the Intermediate Value Theorem but I can't figure out which function to use, since $\alpha, \beta$ are not necessarily of the same/opposite signs.

Answer 1

HINT: Cauchy Mean Value Theorem. Assume that you can parametrize $C$ differentiably by $(f(t),g(t))$, $0\le t\le 1$, with $(f(0),g(0)) = (1,0)$ and $(f(1),g(1)) = (0,1)$. I think the hypotheses on the tangencies at the two points are unneeded.