The six basic formulas of mean inequality are Hn≤Gn≤An≤Qn. 1, mean inequality, also known as mean inequality and mean inequality, is an important formula in mathematics. The content of the formula is Hn≤Gn≤An≤Qn, that is, the harmonic average does not exceed the geometric average, the geometric average does not exceed the arithmetic average, and the arithmetic average does not exceed the square average.

2. There are many methods to prove mean inequality, such as mathematical induction (first mathematical induction or backward induction), Lagrange multiplier method, Qinsheng inequality method, rank inequality method, Cauchy inequality method and so on.

3. The basic formula of the mean value: x, Y ∈ R+, X+Y = S, X Y = P is known. If p is a constant value, then s has a minimum value if and only if x = y；; If S is a constant value, then P has a maximum value if and only if X = Y. Or when A, b∈R+, a+b=k (fixed value), a+b≥2√ab (fixed value) is an equal sign if and only if A = B.

4. Let X 1, X2, X3, ... and Xn be a number greater than 0, then X 1+x2+x3+...+Xn ≥ n times n times x 1 times x2 times x3 times Xn. Mean value theorem, also known as basic inequality. The main content is that in the range of positive real numbers, the geometric mean of some numbers does not exceed their arithmetic mean, and when all these numbers are equal, the arithmetic mean is equal to the geometric mean.

5. The mean value theorem is a very important knowledge point in high school mathematics learning, which is often used in finding the maximum value of a function. The characteristics of the mean value theorem: one positive: all parts are positive numbers. Binary: not equal to left or right is a fixed value. Three phases are equal: an equal sign can be obtained.