so, i am working on some optimization problems and I fully understand the steps on how to find the dimensions that will minimize/maximize the amount of material needed, but i'm having trouble defining these concepts in normal English.
As an example, I will use a standard cylinder problem that gives you the volume and asks to determine how much material you will need to get the maximum/minimum surface area.
As I understanding it, the first derivative tells me that the surface area of the cylinder is changing at a specified rate when the radius is equal to a particular value. This rate of change varies, depending on the measurement of the radius.
It is the second derivative that gives me problems. As I understand it, the second derivative tells me that the rate of change of the surface area's rate of change varies depending on the value of the radius. Therefore, the minimum point of the second derivative tells me that the rate of change of the rate of change is at its least when the radius is a particular value (in graphical terms, the x value of the minimum point).
What I am struggling to understand is how this minimum point of the second derivative influences the amount of material needed for the surface area.
sorry if I overcomplicated this, but the books and classes I use do not do a good job of explaining this at all. they just give you the steps and computation for how to do it, but don't provide real evidence as to how it works or what the information tells you in 'real life' terms.