Given domain and range of a monotone function, what is the maximum slope?

Question

I'm looking for a reference in answer to one of the following questions:

1. Is there a general result out there that will give the maximum possible slope of a monotone function, given its domain and range?
2. Alternatively, is there a specific proof with regard to the cdf of the normal distribution?

I can prove to myself via simulation that the slope of a given normal cdf peaks at a certain value, but I would like to refer to a proved result. Thanks!

Answer 1

Does this work for 2?

CDF of a normal distribution is $\Phi(x)$

The slope at each point is the pdf: $\phi(x)$

Taking the derivative: $\phi'(x) = -x\phi(x)$

Thus, we can see there is a critical point at $x=0$ and the second derivative $-\phi(x) +x^2\phi(x) $ is clearly negative at $x=0$. So the maximum slope of the CDF is $\phi(0) \approx 0.39$

Edit: I suppose this answer can also be generalized to cases where you're given a range of $x$.