Finding the maximum value of basic inequality X > 0. Find the maximum value of X & # 178; + 4 / X!

Finding the maximum value of basic inequality X > 0. Find the maximum value of X & # 178; + 4 / X!

x²+4/x
=The cube root of X & # 178; + 2 / x + 2 / X ≥ 3 (X & # 178; * 2 / X * 2 / x) = the cube root of 3 * 4
Take the equal sign when X & # 178; = 2 / X
So the minimum is the cube root of 3 * 4

If lgx + lgY = 2, then the minimum value of 1x + 1y is______ ．

From lgx + lgY = lgxy = 2, we get xy = 102 = 100, X ＞ 0, y ＞ 0, ｜ 1x + 1y = x + YXY ≥ 2xyxy = 2100100 = 15, if and only if x = y is equal sign, then the minimum value of 1x + 1y is 15

Given that the positive number A.B satisfies 4A + B = 30, so that 1A + 1b takes the minimum, then the real number pair (a, b) is______ ．

∵ if and only if Ba = 4AB, i.e. a = 5, B = 10, the minimum value of 1A + 1b is 0.3. The real number pair (a, b) is (5, 10). So the answer is: (5, 10)

Inequality preserving of function limit
Can we prove the inequality preserving property of lower function limit?

Thinking analysis: it can be seen that the essence of sign preservation is that the function value keeps the same sign as the limit value in a certain range (in a certain change process). To formally prove it, we only need to start from the definition of limit (ε - δ statement), and deduce that the function value is also greater than or less than 0 in the case of a 〉 0 and a & lt; 0