How does the second deriative work i understand the first deriative but i dont really understand how the second deriative gives you the answer of a whether a stationary point is a minimum or maximum?

Answer 1

The first deriviative is the slope of the function. Your understanding is somewhat correct, in that when the first deriviative is zero, the function is at a minimum or a maximum. (Or an inflection point)

Take this one step further. The second deriviative is the slope of the first deriviative. Take a moment to understand what this means...

If the slope of a function is positive, then the value of the function is increasing. If the slope of a function is negative, then the value of the function is decreasing. Last, if the slope of a function is zero, then the value of the function is not changing - this is why a value of zero for the first deriviative means you are at a maximum or a minimum. (Or an inflection point - more on that later.)

OK. If the slope of the first deriviative, i.e. the slope of the slope of the initial function is positive, that means the first derviative is increasing, which means the function is accelerating upward.

Make sure you understand the geometric implications here - it is critical towards understanding the problem. If the first deriviative is zero and the second deriviation is positive, then you are sitting at a minimum. If the first deriviative is zero and the second deriviative is negative, then you are sitting at a maximum.

Now for the inflection point... If the first deriviative is zero and the second deriviative is zero, you are not at a maximum or a minimum - you are at an inflection point - a point where the slope goes to zero - but you don't know if it is going to change to the same direction it is going.

For example, look at y = x3. The first deriviative y' = 3x2, which is zero at x=0, and the second deriviative y'' = 6x, which is also zero at x=0. The point x=0 is an inflection point, not a maximum or minimum, and you know that because the second deriviative is also zero.

You can take this to any level you want, such as looking at the third deriviative. The original question was how to identifiy if a point where the first deriviative is zero represents a maximum or a minimum.