How to find the minimum value of the tick function:

For the form f(x)=x+a/x ("√a" is "a under the root sign"). When x>0, there is a minimum value, which is f(√a); when x=2√ab[a and b are both negative]).

For example: when x>0, f(x) has a minimum value. According to the mean value theorem: x+a/x>=2√(x*a/x)=2√a, so f The minimum value of (x) is 2√a.

Extended information:

The tick function is a general hyperbolic function similar to the inverse proportional function, which is in the form of f(x)=ax+b/x (ab>0 ) function. Common a=b=1. Because the function image is similar to the Nike trademark, it is also known as the "Nike function" or "Nike curve".

The general form of the check mark function is: (x)=ax+b/x(a>0) However, in high school liberal arts mathematics, a is mostly only 1, and the value of b is variable. Changes in science and mathematics are more complex.

The definition domain is (-∞, 0) ∪ (0, +∞) and the value range is (-∞, -2√ab] ∪ [2√ab, +∞). When x>0, there is x=square root b/square root a, the minimum value is 2√ab. When x<0, x=-square root b/square root a, the maximum value is: -2√ab

The analytical formula of the tick function is y=x+a/x (where a>0). The monotonicity of the tick function is discussed as follows: Suppose x1

Function definition

The tick function refers to a function of the form f(x)=ax+b/x (ab>0).

Properties< /p>

Image:

The image of the tick function is two curves with the y-axis and y=ax as asymptotes respectively, and any point on the image reaches the two asymptotes The product of the distance is exactly the product of the sine of the angle between the asymptotes (0~180°) and |b|.

If a>0, b>0, in the first quadrant, the turning point is (√b/a, 2√ab

Maximum value

When the domain is (0~∞), f(x)=ax+b/x(a> 0, b>0) takes the minimum value at x=√b/a, and the minimum value is 2√ab. When the domain is (-∞, 0) ∪ (0, +∞), the function has no maximum value, when When the domain is (-∞, 0), (a>0, b>0) takes the maximum value at f(x)=ax+b/x, x=-√b/a, and the maximum value is -2√ ab.

Odd-even, monotonicity

Parity

The tick function is an odd function.

Monotonicity

< p>Let k=√b/a, then: increasing interval: {x|x≤-k} and {x|x≥k}; decreasing interval: {x|-k≤x<0} and {x|0

Change trend: first increases and then decreases on the left side of the y-axis, first decreases and then increases on the right side of the y-axis.

Asymptote

The two asymptotes of the tick function are the y-axis and y=ax.

Facing this function f(x)=x+b/x, we should think more and need to go deeper. Explore:

(1) What are the applications of its monotonicity and parity, and the range problem is closely related to monotonicity, so the first question that the proposer thinks of should be related to the value range;

(2) There is a close connection between functions and equations, so the proposer will naturally think of the application of the ideas of functions and equations;

(3) As we all know, there are many fixed values ??in the hyperbola problem, so it is easy to think of the existence of fixed values, so the general conclusion is drawn from the special;

(4) Continue to expand and use the results of conjecture and exploration to solve more complex functions. The maximum value problem. Is it related to the mean value?