Do some experiments:

Choose the logistic function. Check the first derivative box. Find an approximate value where the increase of is highest.

Choose the exponential function. Check the second derivative box. Find an approximate , the inflection point, where the tangent line switches from above the function to below. Actually, it is the zero of .

Choose the polynomial of degree four. Check both the first derivative and second derivative boxes.

How big is when has its local maximum and local minimum?