Final answer:
The statement is true; the maximum growth rate for a logistic function occurs when the population size is at half the carrying capacity, leading to an S-shaped logistic growth curve.
Step-by-step explanation:
The statement that for a logistic function the function shows a maximum growth rate at half the limiting value is true. In logistic growth models, the rate of population growth is fastest at the midpoint of the curve, where the population size is at half the carrying capacity (K). This is because at that point, there is a balance between the resources available and the size of the population. As resources start to become more limited with the increase in population, the growth rate slows down. This creates the S-shaped curve that is characteristic of logistic growth, whereas in the absence of limiting factors, a population would grow exponentially - represented by a J-shaped curve. The logistic growth model is important in predicting how populations will grow when confronted with limited resources, and it is visually distinct from exponential growth as it levels off once carrying capacity is reached.