Conjecture: If a line never intersects a differentiable curve then there exists at least one tangent to the curve whose slope is that of the line

Question

This is my conjecture:

If a line never intersects a differentiable curve then there exists at least one tangent to the curve whose slope is that of the line

I have tried proving it by maybe Lagrange's Mean Value Theorem but to no avail...

Answer 1

Not true. Consider the curve $y=e^x$. It never intersects the $x$-axis, but there is no point where the tangent has slope zero.

(No, there's nothing special about the slope being zero: Take the curve $y=ax+e^x$ and the line $y=ax$.)

Answer 2

As you have alredy been informed, the statement is false. However, if we are dealing with a loop or, to be more precise, with a differentiable function $c$ whose domain is $S^1$, then it is true. Just take the point $\theta\in S^1$ such that the distance from $c(\theta)$ to the line is the smallest (or the greatest) possible. The compactness of $S^1$ (and the continuity of $c$) assures us that such a $\theta$ exists. Then the line tangent to the curve at $c(\theta)$ will be parallel to the line. Therefore, they will have the same slope.