Why make the derivative 0 to find the extremum

Because for a derivative function, the derivative of its extreme point must be equal to zero, because the increase and decrease of the two sides of the extreme point must be different, that is to say, the positive and negative derivatives near both sides of the extreme point are different, and the extreme point becomes a transition point, after which the derivative changes from positive to negative or vice versa, and the derivative of most functions is continuous, so the derivative of the extreme point is 0. This understanding is a bit rough, and later you will learn the precise definition of extreme values and some proofs, and then it will be very clear.

For a continuous function, you can also think of it this way, you can draw the curve of the function on the coordinate axes (x,y). When its reciprocal is zero, then the tangent of the point must be parallel to the x-axis (the so-called reciprocal is the tangent of the point on the function. i.e. the derivative is zero).

You can draw a curve and look at it. That point must be an extreme point (maximum or minimum). So to find the extremum, you must first find the point with a 0 derivative.

Of course, a point with a 0 derivative is not necessarily an extreme point. It's college knowledge. For example:

y x3 (cubic ) with a derivative of 0. at (0,0).But it's not an extreme point.

The statement "the derivative at the extreme point is 0" is incorrect.

In general, this is true, because a point with a function derivative of 0 is either a maximum or a minima, but! There are exceptions here, and the second floor is a special case, y

x|This function is in x

0 is a minima, but at x

At 0, there is no derivative for the function.

To sum up, the statement that the derivative at the extreme point is 0 is not true!!

Not exactly.

It should be discussed on a case-by-case basis, because there are cases where extreme points are not derivable.

For example, the function |x|(Draw the picture as a v) takes a minimum at 0, but is not derivative at 0, because the left reciprocal = -1, the right reciprocal = 1, the left reciprocal ≠ the right reciprocal.

The derivative of an extreme point is not necessarily 0. For derivative functions, the image is generally smooth, and the extreme point tangent must be horizontal, that is, the slope of the extreme point tangent is 0, and the extreme point derivative is 0. If the function is monotonionic on both sides of a point with a derivative of 0, then this point is not an extreme point, for example, y=x 3 has a derivative of 0 at x=0, but the function is monotonically increasing on both sides of the origin, and x=0 is not an extreme point.

The concept of extremum comes from the problem of maximum and minimum values in mathematical applications, the maximum and minimum values of the function are collectively called the extremums of the function, and the point at which the function obtains the extreme value is called the forward slip extremum point, and every continuous function on a bounded closed region must reach its maximum and minimum values, and the problem is to determine at which points it reaches the maximum or minimum value. If it is not a boundary point, it must be an interior point, and then this interior point must be an extreme point.

If there is a neighborhood at a point of a function, the function is defined everywhere in that neighborhood, and the function value of the point is the maximum (small), then the value of the function at that point is a maximum (small) value. If it is larger (smaller) than the value of the function at all other points in the neighborhood, it is a strictly maximum (small) value. This point is accordingly referred to as an extreme point or strict extreme point.

For example, the Keira function increases monotonically in the left neighborhood of the extreme point, and decreases monotonically in the right neighborhood of the extreme point.

The derivative of an extreme point is not necessarily zero.

The derivative of an extreme point is not necessarily zero. For derivable functions, the graph image is generally smooth, and the tangent of the extreme point must be horizontal, that is, the slope of the tangent of the extreme point is zero, and the derivative of the extreme point is zero.

If the function is monotonionic on both sides of the point where the derivative is zero, then this point is not an extreme point, for example, y=x 3 has a 0 derivative at x=0, but the function on both sides of the origin of the rift is monotonically increasing, and x=0 is not an extreme point.

The extreme point is the horizontal and vertical coordinates of the maximum or minimum value point in a sub-interval of the function image. If f(a) is the extreme value of the function f(x), then a is said to be the extreme point corresponding to the x-axis when the function f(x) obtains the extremum.

The extreme point is the abscissa of the upper maximum or minimum point in a sub-interval of the function image. Extreme points appear at the stationary point of the function (a point with a 0 derivative) or at a non-derivable point (if the derivative does not exist, an extreme value can also be obtained, in which case the stationary point does not exist).

Explanation of extreme points in academic writing

1. The extreme point of the vertical curveThe highest or lowest point on the vertical curve is called the extreme point.

2. The maximum point and the minimum point are collectively referred to as extreme points. The maximum and minimum points are phased, so the number of extreme points is even. m(p) records the number of extreme points of p.

In addition, it is clear that the feature point must be the inner maximum, and we see that the extreme point of p sequentially divides the edges of p into several chains, each of which is monotonic with respect to y.

1. The derivative refers to the positive hunger derivative function, which is a new function, and if this new function has an extreme value, it is the derivative function.

The extreme value has its own derivative.

2. Generally speaking: "The derivative has an extreme value, and the derivative is not necessarily 0", which is not true.

It should be said: "The derivative function has a pole family without value, and the derivative of the derivative function at its own extreme point must be "

The derivative is 0, which refers to the tangent level of the function, and there are two cases of horizontal tangents:

One is like y=x squared, this function looks like x=0, this is the extreme point;

The other is y=x cube, which looks like this function at x=0, which is called the inflection point;

In addition, the front cavity is empty, and your first half of the sentence is not correct, not the derivatives of the extreme value of the Hui Burning point are all 0, it should be said that the extreme point derivatives of the derivative function are all 0, because the extreme point may also have no derivative, for example, y=|x|In the case of x=0.

You can see it by drawing the image of these three functions yourself and comparing them. , 5, for example: f(x)=x

f'(x)=3x²

When x=0, f'(0)=0.

But f'(x) 0, a circle such as f(x) is increasing on r, at x=0 is not an extreme point. ,1,No**,Understand it this way,The derivative reflects the angle value of the tangent of the graph,The tangent of the extreme point is horizontal,That is, the angle is 0,So,Its derivative is 0。 The derivative of a constant is also 0, and that's because its function graph is a line, without any curvature.

So the derivative of an extreme point is zero, but a point with a zero derivative is not necessarily an extreme point...0,

Answer: First, the first derivative at the extreme point is equal to 0, i.e. f(x).'=0 second derivative f(x).''i.e. the derivative of the first derivative, which is greater than 0, i.e. the first derivative f(x).'is incremental.

So the first derivative f(x) around the extreme point'>0

That is, in the left domain where the first derivative is equal to 0, f(x) is monotonically decreasing, while in the right neighborhood, f(x) is monotonically increasing.

Therefore, it can be seen that the extreme point is the minima!

I suggest you understand the logic well! Deal with f(x) f(x).' f(x)''The relationship between !

For the derivative function (the slope of the tangents of each point on the image exists), the image is smooth, and the tangent of the extreme point must be horizontal, i.e., the slope of the tangent of the extreme point is 0, and the derivative of the extreme point is 0.

If the function is monotonionic on both sides of a point with a derivative of 0, then this point is not an extreme point, for example, y=x 3 has a derivative of 0 at x=0, but the function is monotonically increasing on both sides of the origin, and x=0 is not an extreme point.

e.g. y=|x|The definition of the derivative is that the left derivative = the right derivative and the left and right derivatives of this function are -1 respectively, 1 is not equal, so it does not exist, as in the above formula, at x = 0 minimally add that the derivative = 0 is not necessarily an extremum, and whether it is an extreme value and the derivative is not necessarily related. From the definition of the extreme, the smallest is the nearby one"Small"The neighborhoods are all smaller than the point.

In the case of unary functions:

When the first derivative is zero, the function is an extreme suspicious point. But this is not enough, it also needs to have all odd derivatives zero, such as f(x)=x 3 at x=0. Although f'(0) = 0, but f'''(x)=6≠0, so the function is not an extreme point at x=0, but only an inflection point, which means the point at which the convexity of the function changes.

In multiple situations:

Taking binary as an example, when the first partial derivative of f(x,y) at a certain point is zero, the function is an extreme suspicious point at that point, and the value of the Haysom matrix of the function should be looked at. When the Haysom matrix is positively timed, the function is a minimum at that point, such as the function Z=X 2 2P+Y 2Q at the origin; In negative timing, the function takes a maximum value at that point, such as z=-(x 2 2p+y 2q) at the origin (both function images are paraboloids); The non-extreme value of the timeless function at this point, such as the function z=x 2 2p-y 2q at the origin (the image of the function is the saddle surface, and the saddle point is the origin).

The situation of more than binary is more complicated and inconvenient to explain, so it is recommended to consult relevant books.

Using the definition of a derivative, if a derivative point is an extreme point. Then the derivative must be 0, but if the derivative is 0, it is not necessarily an extreme point. For example, y=x may be an extreme point: a point with a derivative of 0, or a non-derivative.

Therefore, when looking for an extreme value, the first derivative f is usually found first'(x),,req f'(x)=0, solve x, and then determine whether it is an extreme point.