How to judge the second derivative, how to judge the convex, convex and concave concave

I'm a front-line high school math teacher, and I hope I can help you. Take any two points on the image of the function f(x), if the part of the function image between these two points is always below the line segment connecting the two points, then the function is a concave function. Intuitively, the convex function is the image that protrudes upwards.

For example, if the function f(x) is second-order derivable over the interval i, then f(x) is a sufficient and necessary condition for the concave function on the interval i.

It's f''(x)>=0;f(x) is a convex function over interval i, and the sufficient and necessary condition is f''(x)<=0;In layman's terms, if a function finds the first derivative (e.g., greater than o), it can only indicate that it is increasing, but it is not known whether it is increasing faster and slower (it can be analogous to acceleration).

It is only after finding the second derivative that we know the increasing velocity, i.e., convexity.

f"(x) >0: The figure is concave downward.

f"(x) <0: The graph is convex upwards.

Find the first derivative f of the function'(x)、

The second derivative f"(x) if:

f'(x)>0；f"(x) <0: The graph of the function is a curve that increases monotonically on the "convex" ".

f'(x)<0；f"(x) <0: The function graph is a monotonically increasing "down" "convex" curve.

f'(x)>0；f"(x) >0: The graph of the function is a curve that increases monotonically "up" and "concave".

f'(x)<0；f"(x) >0: The graph of the function is a curve that increases monotonically "down" and "concave".

To sum up: f"(x) <0: The figure is convex.

f"(x) >0: The figure is concave.

Extended information: The derivative of the function y=f(x) y = f (x) is still a function of x, then the derivative of y = f (x) is called the second derivative of the function y=f(x). Graphically, it mainly shows the convexity of the function.

If a function f(x) has f in a certain interval i''(x) (i.e., the second derivative) > 0 constant, then for any x,y on the interval i, there is always f(x)+f(y) 2f[(x+y) 2], if there is always f''(x) <0 is true, then the unequal sign of the above equation is reversed.

If a function f(x) has f in a certain interval i''(x) (i.e., the second derivative) is constant > 0, then a line segment connected by any two points on the image of f(x) on the interval i, the function image between the two points is below the line segment, and vice versa is above the line segment.

The first derivative reflects the slope of the function, while the second derivative reflects the speed of the slope change, which is reflected in the image of the function, which is the concave and convex nature of the function. f (x) >0 with the opening up and the function is concave, and f (x) <0 with the opening down and the function is convex.

Intuitive understanding of convexity and concaveness: let the function y=f(x) be continuous on the interval i, if the curve of the function is above the tangent of any point on it, the curve is said to be concave on the interval i; If the curve of a function is below the tangent of any point on it, the curve is said to be convex on interval i.

Steps to determine the bump interval and inflection point of the curve y=f(x):

1. Determine the domain of the function y=f(x);

2. Find the derivative f in the second order"(x)；

3. Find the point that makes the second derivative zero and the point that makes the second derivative non-existent;

4. Judgment or list judgment to determine the concave and convex interval and inflection point of the curve.

The convexity and concave nature of the function is judged by the trend of the tangent rate of change of the function. The rate of change of the tangent of the function is the first derivative, and the derivative of the first derivative is the second derivative, which reflects the change trend of the tangent rate of the function, that is, when the slope of the tangent of the function is positive, negative or zero, the inflection point is determined, so as to judge the convex and concave nature of the function curve. The positive and negative values of the second derivative reflect exactly this situation, so the convexity of the function can be judged.

Because as the bumps change, the tangent slope of the curve changes accordingly.

1 At the lowest point of the concave or the highest point of the convexity, the tangent slope is 0, that is, the first derivative is 02 at the lowest point of the concave image, and the first derivative tends from the lowest left >> to the right <0, and the second derivative >> 0 in this process

Around the top of the convex image, the first derivative tends from the <0 to the right, and the second derivative of this process < 0

Therefore, the concave and convex properties of the function can be judged according to the second derivative.

According to the concave and convex nature of the curve, when f (a) > 0, the curve is concave at point a; f (a) < 0, the curve is concave at point a. If the curve is concave at point A as positive and concave as negative (both are set below), then the positive or negative of the concave direction is the same as the positive or negative of f (a), and the positive or negative of f (a) indicates the positive or negative of the curve concave at point a.

If the first derivative of the function is equal to 0 at that point, and the second derivative is greater than 0, then the graph is concave.

If the function is equal to 0 at the first derivative and the second derivative is less than 0, then the graph is convex.

<> the relationship between the concave and convex properties of the function and the second derivative: the second derivative reflects the speed of the slope change, which is reflected in the image of the function.

Extended Materials. f(x)>0, the opening is upward, the function is concave, and f(x).

Concave. If the second derivative is greater than 0, the first derivative of the function is a single increasing function. That is, the tangent slope of the function at each point increases as x increases. As a result, the function graph is concave.

The second derivative is the derivative of the original function, and the original function is quadratically derived. In general, the derivative of the function y=f(x) y'=f'(x) is still a function of x, then the derivative of y'=f'(x) is called the second derivative of the function y=f(x). Graphically, it mainly shows the convexity of the function.

The speed at which the tangent slope changes is the rate of change of the first derivative. The concave and convex nature of the function (e.g. the direction of acceleration is always directed towards the concave side of the trajectory curve).

If the second derivative is less than 0, the function image is indeed convex, but by definition it is a concave function (the arc of any two points is always above the line connecting these two points).

Conversely, if the second derivative is greater than 0, the function image is concave and by definition a convex function (the arc of any two points is always below the line connecting these two points).

Theorem Let the function y=f(x) be continuous in [a,b] and have first and second derivatives in (a,b), then (1) if in (a,b) and f(x)>0, then the curve y=f(x) is concave on [a,b]. (2) If in (a,b), f(x)<0, then the curve y=f(x) is convex on [a,b].

The relationship between the sign of the second derivative and the convexity of the function

Looking at the tangent of the concave function in the figure below, the slope of the tangent seems to be increasing.

In fact, it is also true that the tangent slope of the concave function increases with the increase of x, and conversely, the tangent slope of the convex function decreases with the increase of x, and the geometric meaning of the second derivative is the slope of the tangent of the image, which corresponds. That is, if the function is a concave function, then the second derivative is greater than 0, and if the function is a convex function, then the second derivative is less than zero.

If the second derivative is greater than 0, the original function is a concave functionLess than 0 is a convex function.

Let f(x) be continuous on [a,b] and have first and second derivatives in (a,b), then:

1) If in (a, b) f''(x) >0, then f(x) on [a,b] is concave.

(2) If in (a, b) f''(x) <0, then f(x) on [a,b] is convex.

Properties of convex functions:

The convex function f defined in a certain open interval c is continuous within c and is differential at all but a countable point. If c is a closed interval, then f may be discontinuous at the endpoint of c.

A unary differentiable function is convex over an interval if and only if its derivative is monotonically undiminished on that interval, a unary continuous differentiable function is convex on an interval if and only if the function is above all its tangents: for all x and y in the interval, there is f(y)>f(x)+f'(x)(y−x）。In particular, if f'(c) = 0, then c is the minimum value of f(x).

If the second derivative is greater than zero, the concave and convex properties of the original function are concave.

If the second derivative is greater than 0, the first derivative of the function is a single increasing function. That is, the tangent slope of the function at each point increases as x increases. As a result, the function graph is concave.

The second derivative is the derivative of the original function, and the original function is quadratically derived. In general, the derivative of the function y=f(x) y'=f'(x) is still a function of x, then the derivative of y'=f'(x) is called the second derivative of the function y=f(x). Graphically, it mainly shows the convexity of the function.

Another expression for the concave and convex nature of the second order derivative greater than 0 is:

a=limδt 0 δv δt = dv dt (i.e., the first derivative of velocity versus time).

And because v=dx dt so there is:

a=dv dt=d x dt is the second derivative of the meta-displacement vs. time.

Applying this idea to functions is what mathematics calls the second derivative.

f'(x)=dy dx (the first derivative of f(x)).

f''(x)=d y dx =d(dy dx) dx (second derivative of f(x)).