Proof: Check function y=x a/x ? (agt; 0)

When xgt; 0, when x=√a, y has a minimum value 2√a

The proof is as follows:

x a/x-2√a

=(√x)?-2√x*√(a/x) [√ (a/x)]?

=[√x-√(a/x)]?≥0

∴ x a/x≥2√a, equal sign when x= It holds when √a

∴ When x=√a, y has a minimum value 2√a

The check mark function is a general hyperbolic function similar to the inverse proportional function, which is of the form Such as the function of f(x)=ax b/x (agt; 0, bgt; 0). Named after the image, it is also known as the "double hook function", "hook function", "check mark function", "double flying swallow function", etc. Because the function image is similar to the Nike trademark, it is also known as the "Nike function" or "Nike curve".

There is a rule here: if and only if ax=b/x, get the minimum value, solve for x=sqrt(b/a), and the corresponding f(x)=2sqrt(ab). Let’s take a look at the mean inequality again. It can also be written like this: (a b)/2≥sqrt(ab). As everyone knows, the previous formula is the formula for finding the average. What about the following formula? It is also the formula of the average, but the difference is that the former is called the arithmetic mean, while the latter is called the geometric mean. To sum up, the arithmetic mean will never be less than the geometric mean.

The above research is all based on xgt; 0, but the tick function is an odd function, so after studying the properties of the positive semi-axis image, a symmetrical image can naturally be made up. If there is a translation problem (the image is no longer regular), first use the translation formula or the translation rules I summarized to restore it and then study it later. This ability is very important. You must practice more and strive to become particularly proficient.