Cauchy inequality was obtained when Cauchy, a great mathematician, studied the problem of "flow number" in mathematical analysis. But historically, this inequality should be called Cauchy-Bunyakovski-Schwartz inequality, because it was the latter two mathematicians who independently extended it in integral calculus that made this inequality applied to a nearly perfect degree.

Direct application of Cauchy inequality

Example: Given that X and Y satisfy x+3y=4, find the minimum value of 4x2+y2.

Analysis:

Method one, the direct thought of everyone after seeing this problem may be to change the binary variable about X and Y into a univariate variable about X or Y, and then solve it with the help of the idea of quadratic function.

Method 2, because its structural characteristics are very similar to Cauchy inequality.