What's the difference between using the size of the second derivative and the left and right sign of the first derivative to judge the extreme value of a function? Sometimes I use the left and right symbols of the first derivative to judge whether it is a maximum or a minimum. The answer is usually the size of the second derivative. Is there any difference between the two methods

What's the difference between using the size of the second derivative and the left and right sign of the first derivative to judge the extreme value of a function? Sometimes I use the left and right symbols of the first derivative to judge whether it is a maximum or a minimum. The answer is usually the size of the second derivative. Is there any difference between the two methods

1. If the second derivative can be used to judge, then the sign of the first derivative can also be used to judge (unless it is difficult to judge the sign of the first derivative of this function). If you say you are wrong, you must be wrong;
2. The difference between the two methods: Generally speaking, if the second derivative is easy to find, it is easier to use the second derivative to judge, but the premise of this method is that the second derivative must exist and not be 0. If the second derivative does not exist or equal to 0, it is still necessary to use the sign of the first derivative to judge. Therefore, the application of the second derivative method is narrower
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Seeking extremum with second derivative
When the value of the second derivative at a certain point is 0, how to judge whether the point is an extreme point?

We should continue to judge whether the first derivative is zero or not. If it is not zero, it is not the extreme point. If it is zero, we should judge whether the positive and negative of the second reciprocal is the same and cannot be zero (if it is zero, the first order will continue to be zero). If it is the same, it is the extreme point

The point where the first derivative is zero and the second derivative is zero in the domain of definition must not be an extreme point, right?
(1) Why not?
(2) If the first derivative in the domain is zero and the second derivative is not zero, then it must be an extreme point. Is this proposition correct? Why?

(1) Y = x ^ 3, at 0, the first derivative and the second derivative are both equal to 0, but 0 is not its extreme point
(obviously, it is not the maximum / minimum value in any neighborhood of 0)
(2) If the second derivative is not zero, it means that the sign of the first derivative changes near the point, so it must be an extreme point
(if the second derivative is greater than 0, the first derivative always increases near this point, and the first derivative is equal to 0 at this point,
So there must be a first derivative 0 on the left side of the point, then it is obviously the extreme point.)

Why can we use the second derivative to find the extremum of a function?

The second derivative is the derivative of the first derivative. Therefore, if the value of the first derivative at a certain point is 0 (indicating that its tangent is a horizontal line), and the derivative value in a monotone interval adjacent to it on both sides is different (the derivative increases monotonically when it is positive and decreases monotonically when it is negative), it can be seen that if the value of the first derivative is 0, the second derivative is a minimum point, Less than 0 is the maximum point