Application of the inverse function thoerem

Question

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a $C^2(\mathbb{R})$ function whose second derivative is bounded away from zero, i.e. $\exists \ c_1, c_2 > 0$ such that $0 < c_1 < |f''(x)| < c_2 \ \forall x\in\mathbb{R} $. Moreover, suppose $f'$ has image $\mathbb{R}$. Then $f'$ is invertible. Define $l:\mathbb{R}\rightarrow\mathbb{R}$ as the inverse of $-f'(x)$.

Question: What is $l'(y)$?

By the inverse function theorem I get $$l'(y)=\dfrac{-1}{f''(l(y))}$$

However, this differs from what I expected to get by a factor of $-1$. Why should $l'(y)$ be positive?

Answer 1

Since $f''$ is continuous and bounded away from zero, there are two possibilities:

1. $f''(x)>0$ for all $x\in\Bbb R$
2. $f''(x)<0$ for all $x\in\Bbb R$

In the first case $f'$ is increasing, $-f'$ is decreasing, so is $l$ and $l'$ is negative. In the second case, the other way around.