The Second Derivative - Inflection Points
The second derivative is well, the derivative of the first derivative of a function.
Essentially, it's the rate of change of the rate of change of a function.
Suppose you had a function, which you knew nothing about, and were to study it at a point. All you know is that it's always been positive up until that point, at which it's zero. Somehow, its derivative is also 0. But you are given that its second derivative is positive.
Think about it. Whatever this function is, it will be going down until it's zero at that point. However, at that point, the first derivative is also zero. Oh no!
This is where the second derivative comes in. Given that it's positive, past that point, however little, the first derivative will increase, and so will the original function.
This means that it will be positive in a neighborhood around that point.
Let's talk about concavity.
A function can be concave or convex. But what does that actually mean?
We talked about critical points. They're points where the first derivative is zero, around which the derivative switches signs. If this function goes back down, or back up, that point is a critical point.
But what happens when it continues to go back up?
Consider
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The function, its derivative, and second derivative are all 0 at x = 0. However, we know that the third derivative is 6. So, as we pass 0, even if for an infinitely small range, the second derivative will be positive, thus also the first and therefore the original function.
Notice that this function goes from a "half parabola" pointing down to a "half parabola" pointing up.
This "half parabola" pointing down is when the function is concave.
Similarly when this "half parabola" points up, the function is convex.
A function is convex over an interval if the second derivative is positive over that interval and concave over an interval if the second derivative is negative over that interval.
The point where the second derivative is zero and the function switches concavity, so, the second derivative changes sign, is called an Inflection Point of that function.
In another way of explaining:
If you have a line tangent to the graph of a function, if the function is convex, the graph of the function will always be above that line.
Similarly, if the function is concave, the graph of the function will always be below that line.
Done with this one!