摘要
本文推广了微积分学中的牛顿—莱不尼兹公式,并利用它给出凸函数可导性的又一个证明。
The main results of this paper are the following
Theorem 1. Let f(x) be a continuous function on [a,b]. If for all x in (a,b) at least one of the three derivatives f_+ (x), f_- (x) and f^s(x) exists and is nonnegative, then f(x) is monotonic increasing.
Theorem 2. Let f(x) be Riemann integrable on [a,b] and let g(x) be continuous on [a,b]. If for all x in (a,b) at least one of the three derivatives g′_+ (x), g′_- (x) and g′(x) exists and is equal to f(x), then
integral from n=a to ∞ b f(x)dx=g(b)-g(a)
关键词
微积分
凸函数
牛顿
莱布尼兹
可导
One—sided derivative, symmertric derivative, fundamental theorem of calculus, convex function.
