Definition method, known conclusion method and second derivative method of function can be used to determine the method. For convex functions on real number sets, the general judgment method is to find their second derivative. If its second derivative is nonnegative in the interval, it is called a convex function. If its second derivative is always greater than 0 in the interval, it is called a strictly convex function.
If the derivative f' of the differentiable function f rises monotonously in a certain interval, that is, the second derivative exists, then in this interval, the second derivative is greater than zero and f is concave; That is to say, a concave function has a declining slope (where declining only means not rising but strictly falling), which means that it allows the existence of zero slope. )
If the second derivative f'(x) of quadratic differentiable function f is positive (or it has positive acceleration), then its image is concave; If the second derivative f'(x) is negative, the image will be convex. If a certain point changes the convexity of the image, it is an inflection point.
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Any minimum of a convex function is also a minimum. A strictly convex function has at most one minimum value.
For convex function f, horizontal subset {x | f (x)
Jensen inequality holds for every convex function f, if x is a random variable and takes a value in the definition domain of f, then (here e stands for mathematical expectation. )
Another important property of convex functions is that for convex functions, local minima are global minima.
Baidu Encyclopedia-Convex Function
Baidu encyclopedia-concave function