optimization

Hello

I have no idea how to solve this:

Find the first-order condition and the second-order condition and characterize the ekstreme point/extreme points for the function:

where

Please explain it to me very slowly and carefully, so that I can understand what in the world I'm supposed to do here

I'm sorry, but I'm still pretty confused?

No one to help?

An extreme point of a function is when the function reaches either its highest or lowest value.You can find this information out by finding the derivative. This gives you the gradient of the function, in other words, it tells you, if I change x by a minuscule amount, how much does y change (assume y = f(x) from here on in)?

The reason the gradient can help you find these "minimum" or "maximum" points is that when the gradient is zero, its telling you that the equation, with a small change in x, isn't changing in y anymore (we're talking infinitesimal amounts here). Or better worded, the rate of change at that maximum or minimum point is zero. The function isn't changing with small changes.

So the first step will be to find the derivatives. The second step will be to set those derivatives to zero. Finally, solve for x,y, which will be the co-ordinates on the equation where the curve is a maximum or minimum.

The second order derivative simply means finding a derivative of another derivative. It is exactly like acceleration, velocity and distance. Here, distance is x, the derivative with time, is velocity (how is the distance changing with time). But the second order derivative is asking how how is the velocity changing with time, or how is that gradient changing with a small change in x. In essence, from arguments before, it tells you whether you have a maximum, or a minimum. So the first order derivatives tell you that you have some maximum and minimum points (if they exist). The second order derivatives tell you whether its a maximum or a minimum.

The reason why all this is important, not only mathematically as described above, but physically, is that very often we are interested in systems in say, the lowest energy state or even just simple kinetics like finding accelerations or velocities.

Please note, the above isn't a rigorous explanation, just a feel for whats going on, at least how Interpret it.