The derivative of logistic growth of a population over time is $$\frac{dP}{dt} = 5P - 0.002P^2$$
When I take the second derivative I end up with $5 - 0.004P$.
This means for populations above 1250, the curve is concave down, but below 1250 the curve is concave up. However above the carrying capacity (2500), the curve should be concave up.
Why is the second derivative wrong in showing that for populations above 1250, the graph is concave down?