Does the maximum value have to be greater than the minimum value?

Not necessarily. Maxima with.

Minimum. It is defined within the domain, that is, in.

Extreme points. Within a very short distance, it is the maximum or the minimum, but in the entirety.

Domain. , it is not the maximum, and there may be a minimum that is larger than the maximum. Extremums are only for the domain, not for the entire defined domain.

For example: Suppose one.

Continuous functions. f(x), the extreme value is f'(x)=0 at the same time as f''(x) A point greater than 0 is a minimum, and a point less than 0 is a maximum. It is this illustration that you can see that the 4 inflection points on the graph are the extreme points, and you can see that the value of the second point on the left (the minimum point) is greater than the value of the rightmost point (the maximum value).

First of all, you have to understand the concept of extremum, which is defined in the textbook, and understand that it is different from the maximum and minimum values. The loose understanding of extreme values is that when the derivative of a continuous function is zero, x=? The corresponding value, from the diagram, is the peak and trough of the wavy line, so let's compare, if a wave has 2 peaks, 3 troughs, and a trough in the middle of the period is higher than a peak in the middle of the period, like a wave, I won't draw a picture, so that the maximum is less than the minimum.

The magnitude of the maximum and minimum of a function is uncertain.

It is enough to distinguish between the maximum and maximum, and between the minimum and the minimum.

No, there are many times in the function that the maximum value is less than the minimum value, you can draw a graph, (a few more arcs) look at it, here I am not good at drawing,

We're also studying this chapter, and in the same function, the maximum value must be greater than the minimum value, otherwise it would not be the maximum value.

Not necessarily, the maximum or small value is simply the largest or smallest value in a range.

Yes, it must be greater than the minimum, I just asked the teacher a few days ago.

Not necessarily, for example, y=1 is a function with a maximum = a minimum.

This is not necessarily, the smallest is still equal, such as 1

Not necessarily, but it may also be equal.

Judgment of maxima and minimum:For functions, increasing first and then decreasing produces a maximum, and decreasing first and then increasing produces a minimum; For the derivative function, negative first and then positive produce maximum, and positive and then negative produce minimum. Within a given interval, there can be multiple maxima and minima, with the largest being the maximum and the smallest being the minimum.

Let x0 be the (local) extremum of f(x) and the derivative of f(x) exists, then the derivative of f(x) is 0, but a zero derivative of f(x) does not mean that x0 is an extremum. To put it simply, if it is a closed interval, then the minimum (maximum) value can be taken on this closed interval, then it is called the minimum value (maximum).

However, if the interval is open, the minimum value (maximum) cannot be obtained, so the concept of derivative is introduced to define the minimum value (maximum).

Introduction

Extremum is a basic concept of variational methods. The maximum or minimum value obtained by the functional within a certain range of the allowable function is called the maximum or minimum value, respectively, and is collectively referred to as the extreme value. The variable function that brings the functional to an extreme value is called an extreme function, and if it is a univariate function, it is often called an extreme curve.

Extremum is also known as relative extremum or local extremum.

Extremum is a collective term for "maximum" and "minimum". If the value of a function at a point is greater than or equal to the value of the function at any other point near that point, the value of the function at that point is said to be the "maximum" of the function. If the value of a function at a point is less than or equal to the value of the function at any other point near that point, the value of the function at that point is said to be the "minima" of the function.

The above content reference: Encyclopedia - Extremum.

For functions, increasing first and then decreasing produces a maximum, and decreasing first and then increasing produces a minimum;

In a very small interval of the function, there are independent variables with the value x, and there are independent variables larger and smaller than it, and the corresponding function values of these independent variables are less than the function values corresponding to x. Then the value of this function is called the maximum, i.e. if all x's in a point x0 have f(x).

Determine whether it is a maximum or a minimum:

Find the second derivative of the function, substitute the extreme value point, the second derivative value "0 is the minimum value point, and vice versa, the second derivative value = 0, it may not be the extreme value point.

Determine the positive and negative values of the derivatives of the neighborhoods left and right of the extreme point: left + right - is the maximum point, left - right + is the minimum point, the left and right plus or minus are unchanged, not the extreme point.

The difference between a stationary point and an extreme point.

The extreme point of a derivative function must be its station, but conversely, the stationing point of a function is not necessarily an extreme point.

Functional: 1An extreme point is not necessarily a stationary point. e.g. y=|x|, which is not derivable at x=0, is not a stationary point, but a point of extreme (small) value.

2.A station is not necessarily an extreme point. For example, y=x, the derivative is 0 at x=0, which is a stationary point, but there is no extreme value, so it is not an extreme point.

1. Steps to find the maximum minimum:

Find the derivative f'(x)；

Find equation f'(x)=0;

Check f'(x) the sign of the value on the left and right of the equation, if the left is positive and the right is negative, then f(x) takes the maximum value at this root; If the left is negative and the right is positive, then f(x) takes a minimum at this root.

f'(x) Pointless points should also be discussed. You can find f first'(x)=0 and f'(x) Meaningless points, and then judge them according to definitions.

2. Steps to find extreme points:

Find f'(x)=0,f"(x) the value of ≠0;

The discontinuity of f(x) is discussed by the definition of an extreme value (the point where the radius of the neighborhood f(x) is smaller or larger than the point is the extreme point).

The set of all the above points is the set of extreme points.

1. Different definitions.

1. Extreme point: If f(a) is the maximum or minimum value of the function f(x), then a is the extreme point of the function f(x), and the maximum point and the minimum point are collectively referred to as the extreme point. 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).

2. Station: The first derivative of the function is 0 (stationary point is also called stable point, critical point). For multivariate functions, a stationary point is a point where all first-order partial derivatives are zero.

3. Inflection point: also known as the recurve point, mathematically refers to the point that changes the upward or downward direction of the curve, and intuitively speaking, the inflection point is the point where the tangent line crosses the curve (that is, the dividing point between the concave arc and the convex arc of the continuous curve).

Second, the nature is different.

1. The monotonicity may change at the station, and the concave and convex may change at the inflection point.

2. Inflection point: the point at which the concave and convex properties of the function change.

3. Stationary: The first derivative is zero.

Third, the characteristics are different.

1. The extreme point is not necessarily a stationary point. e.g. y=|x|, which is not derivable at x=0, is not a stationary point, but a point of extreme (small) value.

2. The stationing point is not necessarily an extreme point. For example, y=x, the derivative is 0 at x=0, which is a stationary point, but there is no extreme value, so it is not an extreme point.

3. The function of the curve graph has a second derivative at the inflection point, then the second derivative has a different sign (from positive to negative or from negative to positive) or does not exist at the inflection point.

Not necessarily. The maximum value is not necessarily greater than the minimum value. Because there are specific domains for the definition of maxima and minimum, the maxima and minima are not necessarily equal in different domains.

This may be the maximum or minimum in a certain region, but this may not be the case in the entire defined domain, so the maximum and minimum are only local.

The difference between maximum and maximum.

The maximum value is the largest value in the function, while the maximum value is not.

The maximum value must be higher than the other values in the function, and the maximum value can be less than the minimum value.

There is only one maximum value, while there can be an infinite number of maximum values.

The interval of the maximum value is defined by the function definition domain, and the maximum value can be defined in the interval of the function.

It is not necessary that the maximum and minimum values are defined in the domain, that is, they are the maximum or minimum values in a very short distance around the extreme points, but in the whole defined domain, it is not the maximum value point, and there may be a minimum value that is larger than the maximum value. Extremums are only for the domain, not for the entire defined domain.

For example, if we assume a continuous function f(x), the extreme value is f'(x)=0 at the same time as f''(x) A point greater than 0 is a minimum, and a point less than 0 is a maximum. It is this illustration that you can see that the 4 inflection points on the graph are the extreme points, and you can see that the value of the second point on the left (the minimum point) is greater than the value of the rightmost point (the maximum value).

The maxima and minima of the function are local, and it is entirely possible that the maximum is less than the minimum. You get the idea. The function obtains a maximum at point A and a minimum at point B, but the minimum at point B is greater than the maximum at point A.

It doesn't have to be as large as a constant function maximum

In a function graph, to determine whether the value is a maximum or a minimum, you need to determine the derivatives sign on both sides of the graph. If the early derivative of hail on the left is negative and the right derivative is positive, then the value of the hail at the place is minimal. If the left derivative is positive and the right derivative is negative, then the value is maximum.

As in the diagram I drawn.

x = a, c, e are maximums and minimals. In addition, since the function value of point e is the largest, it is also the maximum; The value of the function at point d is the smallest and the smallest.

In general, for the original function, first decreasing and then increasing produces a minimum, and increasing first and then decreasing produces a maximum;

Minimum.