Advanced Mathematics Extrema and inflection points of derivative functions

Your questions are basically conceptual ones, and if you look closely at the textbook, you shouldn't have a problem. I'll give you a brief distinction and explanation:

First of all, the extreme point is a local property of a function, specifically if the value of the function at this point is compared with other values in a small neighborhood of this point, the maximum or minimum value is taken, and the corresponding maximum and minimum values are corresponding. This concept has nothing to do with the derivability of the function itself. But for general differentiable functions, a point with zero first derivative is often an extreme point, but it is not absolute, for example, f(x)=x 3, x=0 is not an extreme point.

Generally we put f'A point of =0 is called a stationary point, and there are only two cases of an extreme point, either a stationary point or a non-derivable point. On the contrary, it is not true, and the non-leading or stationary point is not necessarily an extreme point.

Secondly, the inflection point is the point where the convexity and concave properties of the function image (called convex and convex in some textbooks) change, so it is called the inflection point, which has no essential relationship with the extreme point, and reflects two different mathematical properties. Similar to extreme points, inflection points are made up of two types of points: points where the second derivative is zero, and points where the second derivative does not exist.

These are two concepts that can be easily confused.

1) If the function is not derivative at this point, then the extreme point and the inflection point can be the same, such as the piecewise function

When x<0, f(x)=x2;

When x 0, f(x) = x

At x=0 is both an extreme point and an inflection point.

2) If the function is derivable, then the inflection point must not be an extreme point. The way to determine whether it is an extreme point or an inflection point, you only need to look at its 1st, 2nd, and 3rd orders. nth derivative, see which derivative is not 0, assuming that it is not 0 until the nth order, and the first n-1 order is 0, then if n is an odd number, this is the inflection point; If n is an even number, this is the extreme point.

There is no necessary connection between the extreme point and the inflection point.

For example, y=x 3, (0,0) is the inflection point, but x=0 is not an extreme point.

y=x 4, (0,0) is not the inflection point, and x=0 is the minimum point.

When a point on the graph of a function makes the second derivative of the function zero, and the third derivative is not zero, this point is the inflection point of the function.

The extreme point is the abscissa of the maximum or minimum point in a sub-interval of the function image. Extreme points must occur at the stationary point of the function (the point with a 0 derivative) or at the non-derivable point.

The inflection point is not necessarily an extreme point, but the extreme point must be an inflection point.

The first derivative at the extreme point is 0, and the first derivative describes the increase or decrease of the original function; The second derivative at the inflection point is 0, and the second derivative also describes the increase or decrease of the original function.

If the function exists at that point and its domain with a first-order, second-order, and third-order derivative, then the first derivative of the function is 0, and the point where the second-order derivative is not 0 is an extreme point; The second derivative of the function is 0, and the point where the third derivative is not 0 is the inflection point. For example, y=x 4, x=0 are extreme points but not inflection points. If there is no derivative for the point, a practical judgment is required, such as y=|x|, x=0 does not exist when the derivative does not exist, but x=0 is the minimum point of the function.

This is not a canonical textbook, and the concept of "derivative with sufficient order" is made up by young teachers with insufficient teaching experience, and it should be "derivability with sufficient order". Mature older teachers need to be able to withstand nitpicking.

If the second derivative is continuous, or if it is third-order derivable, then the inflection point of [f(x) is f'(x)] The conclusion is true.

To prove this conclusion, there is no need to kill chickens, and the Taylor formula is not used at all.

Use the Lagrangian median value theorem f'(x)-f"(x0)=f"( )x-x0).

f"( ) in the left and right neighborhoods, x-x0 in the left and right neighborhoods, f'(x)-f"(x0)=f"( )x-x0) will not change the sign, and the conclusion will be proven.

Mountain road water bridge.

No. Inflection point: The dividing point between the concave and convex arcs of the continuous curve, and the value of the second derivative function at the inflection point is 0. Explain that the two sides of the inflection point must be a concave arc and a convex arc.

The sign of the second-order derivative function can determine the concave-convex arc of the function, so the second-order derivative of the commotion function must be found first.

The return refers to all points where the value of the second-order derivative is 0;

Then determine the sign of the second-order derivative function value around these points, if the left and right signs are opposite and missing, then the point is the inflection point. Otherwise, it's not.

Wrong. The former is only a necessary condition for the latter, and it may not be sufficient.

First of all, the condition only says that f is derivable, not that f is second-order derivable. It is possible that f takes the maximum value at x0, f'(x0) = 0, but f''(x0) does not exist. For example, if the function f(x)=(sgnx-2)*x 2 is at 0.

Second, even if f is second-order derivable, as you suggest, it is possible that f takes a maximum value at x0, and f'(x0)=f''(x0)=0. For example, the function f(x)=-x 4 is at x=0.

When f'(x0)=f''(x0)=0, if f has a higher derivative at x0, there is a standard discriminant (this may be required by lz):

Denote the nth derivative of f at x0 as f n(x0), if f'(x0)=f''(x0)=…=f_k(x0)=0, f_(k+1)(x0)≠0。Rule.

1) When k is an even number, x0 is not an extreme point;

2) When k is an odd number, x0 is the maximum point if and only if f (k+1)(x0) <0

Note that the premise in (2) is f (k+1) (x0) ≠0.

As for the proof, it is sufficient to use the Taylor formula with the Peano remainder (spread to the (K+1) order).

Although the above discrimination method is sufficient in most cases, it does not solve all cases! If f'(x0)=f''(x0)=…=f k(x0)=0 but f (k+1)(x0) does not exist, and this discriminant method naturally fails.

In addition, even if f is derivable at any time in the x0 neighborhood, it is ......embarrassing if all derivatives of f at x0 are 0Note that this is possible in the case of real functions, where the fact that the derivatives of f at x0 are 0 does not mean that f is a constant function in a neighborhood of x0. The classic example is f(x)=exp(-1 x 2), and f(0) is defined as 0For such a function, the above discriminant also fails.

In summary, adequacy is not right. Extreme points can be judged by the derivative value of this point in most cases, but the derivative does not work in some strange situations......At this point, we can only consider starting with the definition. For example, if it can be shown that there is t>0, x0-t as for the inflection point, it can generally be discussed as an extreme point. In fact, if f is derivable within the x0 neighborhood, then "x0 is the inflection point of f" is equivalent to "x0 is f."'of extreme points". Therefore, similar to the extreme point, it is difficult to say that the inflection point has a simple sufficient and necessary condition, but the above discriminant method is still a good criterion.

In addition, the inflection point is first looked at the second derivative (discuss f'There is no direct connection with F's station!

Actually, it's not hard to prove (lz is interested, it's a good exercise): if a derivable point of f is both a strict extreme point and an inflection point, then f can only be a constant function in one neighborhood of this point! If the word "strict" is removed, the conclusion becomes:

is a constant function in one of the left or right neighbors at this point.

Let me give you a brief answer. f'(x)=0 is called a station; f''The point where (x)=0 is called the inflection point; f'Determines the direction of the curve (determines the increase or decrease of the function in a certain segment), f''Decide on the direction of the opening (perhaps it is more appropriate to call it convexity, but the direction of the opening is easy to understand).

For example, the function is at a certain point f'(x0)=0, cut f'(x)<0 , when x0, when x>x0; Then the graph can tell that xo is the minimum point (similar to the maximum point).

If you have to use the second derivative, then the conclusion is as follows: The function is at a certain point f'(x0)=0, f''(x0)<0 (opening down at x0), so this point is the maximum point.

f'(x0)=0 and f''(x0)<0] I assume that your x0 does not appear on the boundary (e.g. a in the interval [a,b] is an boundary, and if it appears on the boundary, there is only a left derivative or a right reciprocal at that point), and you have assumed that your function is derivative, then we will judge [f'(x0)=0 and f''(x0)<0] can be deduced x0 is the maximum point.

You can refer to Mathematical Analysis, Advanced Mathematics, etc.

In summary, for the derivative function, the "sufficient and necessary condition" where x0 is the maximum point is [f'(x0)=0 and f''(x0)<0】This conclusion is correct.

I'll make it simple for you.

f'(x)=0 is called a station;

f''The point where (x)=0 is called the inflection point;

f'Determines the direction of the curve (determines the increase or decrease of the function in a certain segment), f''Decide on the opening square and cavity direction (perhaps it is more appropriate to call it convex and concave, but the opening direction is easy to understand).

For example, the function is at a certain point f'(x0)=0, cut f'(x)<0

when x0, when x>x0; Then the graph can tell that xo is the minimum point (similar to the maximum point).

If you have to judge by the second derivative, then the conclusion is as follows:

The function is at a certain point f'(x0)=0,f''(x0)<0

at x0 point opening down), so the point is the maximum point.

f'(x0)=0 and f''(x0)<0】

I'm assuming that your x0 doesn't appear on a boundary (e.g., a in the [a,b] interval is a boundary, and if it appears on a boundary, there is only a left derivative or a right reciprocal at that point), and you've assumed that your function is derivative, then [f'(x0)=0 and f''(x0)<0】

It is possible to push out x0 is the maximum point.

You can refer to:

Mathematical Analysis, Advanced Mathematics, etc.

In summary, for the derivative function, the "sufficient and necessary condition" where x0 is the maximum point is [f'(x0)=0 and f''(x0)<0】This conclusion is correct.

Inflection and extreme points are usually not the same.

As you said, the definitions of the two are different.

The first derivative at the extreme point is 0, the first derivative describes the increase or decrease of the original function, the second derivative at the inflection point is 0, and the second derivative describes the concave and convex nature of the original function.

Not really.

From the image.

Look, the inflection point is the dividing point of the concave and convex function image; It can be determined by the second derivative!

The inflection point is the point at which the upward or downward direction of the curve is changed, and intuitively speaking, the inflection point is the point at which the tangent crosses the curve (i.e., the demarcation point of the curve).

If the function of the curve graph has a second derivative at the inflection point, then the second derivative is different at the inflection point (from positive to negative or from negative to positive).