Second derivative joke

The second derivative greater than zero is concave. The second derivative reflects the speed of slope change and is reflected in the image of the function.

The second derivative is greater than 0, indicating that the first derivative of this function is a single increasing function. That is to say, the tangent slope of the function at each point increases with the increase of x, so the function graph is concave. The second derivative is the derivative of the original function, and the original function is derived twice.

The derivative y' = f' x of the general function y=f(x) is still a function of x, so the derivative y'=f'(x) is called the second derivative of the function y=f(x). The graph mainly shows the concavity and convexity of the function, the changing speed of the tangent slope, the changing rate of the first derivative and the concavity and convexity of the function (for example, the direction of acceleration always points to the concave side of the trajectory curve).

The concavity and convexity of that function, let f(x) be continuous in [a, b] and have first and second derivative in (a, b), then:

1, if in (a, b) f'' (x) >; 0, then the graph of f(x) on [a, b] is concave;

2. If in (a, b) f'' (x)

Extended data:

1, if a function f(x) has f''(x) (that is, the second derivative) in an interval I >; 0 is a constant, so for any x and y on the interval I, there is always: f(x)+f(y)≥2f[(x+y)/2], if there is always f'' (x).

Geometric intuitive explanation: if a function f(x) has f''(x) (that is, the second derivative) in an interval I >: 0, then the line segment connected by any two points on the image of f(x) in the interval I, the function image between these two points is below the line segment, and vice versa.

2. Determine the maximum and minimum values of the function.

The extreme value of a function can be obtained by combining the first derivative and the second derivative. When the first derivative is equal to 0 and the second derivative is greater than 0, it is a minimum point. When the first derivative is equal to 0 and the second derivative is less than 0, it is the maximum point; When both the first derivative and the second derivative are equal to 0, it is the stagnation point.

Baidu Encyclopedia-Secondary Derivation