Understanding 2nd Derivative Max/Min Points

[devious_]

I've always wondered why a negative second derivative indicates a maximum point, and a positive one indicates a minimum.

I figured this was because a second derivative is the rate of change of gradient, and because near the maximum point the gradient becomes negative, and vice versa. Am I right?

 

[NateTG]

I've always wondered why a negative second derivative indicates a maximum point, and a positive one indicates a minimum.

I figured this was because a second derivative is the rate of change of gradient, and because near the maximum point the gradient becomes negative, and vice versa. Am I right?

I don't think you're using the usual mathematical notion of gradient.

Consider that the derivative is related to the slope, and that the sign of the second derivative affects the sign of the derivative if it just hit zero.

 

[devious_]

That's what I was trying to say.

At the maximum point, the gradient decreases until it becomes zero (negative gradient). At the minimum point, the gradient increases until it becomes zero (positive gradient).

 

[Zorodius]

Just to state it a slightly different way:

The first derivative of a function gives the slope of that function at any point.
The second derivative of a function tells you how the first derivative (the slope) changes as you move to the right.

If the second derivative is negative, the slope of the function becomes more negative as you move to the right. It is bending downward.

If the second derivative is positive, the slope of the function becomes more positive as you move to the right. It is bending upward.

For example, suppose you have some function, and you know you're looking at either a maximum or a minimum, but you don't know which. However, you see that the function is bending upward there. In that case, it has already gone as low as it can. That means it's a minimum.

If I can slip a question in here, directed at anyone who knows the answer: How, if at all, does the gradient of a single-variable function differ from the derivative of that function? Offhand, I'd think the only difference is that the gradient is considered to be a vector quantity, but since it's a one-dimensional vector, that still might be the exact same thing.

 

[gravenewworld]

We always just found mins and maxes by using the 1st derivative. The 1st deriv. tells you the slope of the tangent line at a specific point. You can find the mins and maxes by setting the slope=0 of the tangent line. From there all you have to do is test points to the left and right of the critical points. If the derivative goes from + 0 - then you have a maximum, or - 0 + you have a minimum at the critical point. This obviously makes sense intuitively.

 

[Zorodius]

We always just found mins and maxes by using the 1st derivative. The 1st deriv. tells you the slope of the tangent line at a specific point. You can find the mins and maxes by setting the slope=0 of the tangent line. From there all you have to do is test points to the left and right of the critical points. If the derivative goes from + 0 - then you have a maximum, or - 0 + you have a minimum at the critical point. This obviously makes sense intuitively.

The second derivative test is easier to calculuate, especially where the function is changing slope rapidly.