②(x ^ n)' = NX(n- 1)(n∈q) 1。 Monotonicity of functions.

(1) Use the sign of the derivative to judge the increase or decrease of the function.

Using the sign of derivative to judge the increase or decrease of function is an application of the geometric meaning of derivative in studying the law of curve change, which fully embodies the idea of combining numbers with shapes.

Generally speaking, in a certain interval (a, b), if > 0, then the function y=f(x) increases monotonically in this interval; If < 0, the function y=f(x) monotonically decreases in this interval.

F(x) is a constant function, if it is always =0 in a certain interval.

Note: In a certain interval, > 0 is f(x), which is a sufficient condition for increasing function, but not a necessary condition. For example, f(x)=x3 is increasing function, but.

(2) the step of finding the monotone interval of the function

① Determine the domain of f(x);

② Deduction;

(3) Use (or) to solve the range corresponding to x. At that time, f(x) was increasing function in the corresponding interval; At that time, f(x) was a decreasing function in the corresponding interval.

2. Extreme value of function

Determination of Extreme Value of (1) Function

(1) If both sides have the same sign, it is not the extreme point of f(x);

(2) if it is near the left and right, it is the maximum or minimum.

3. Steps to find the extreme value of the function

(1) Determine the functional domain;

② Deduction;

③ Find all the stationary points in the definition domain, that is, find all the real roots of the equation;

(4) Check the symbols around the stagnation point. If Zuo Zheng is negative to the right, then f(x) gets the maximum at this root; If the left is negative and the right is positive, then f(x) takes the minimum value at this root.

4. The maximum value of the function

(1) If the maximum value (or minimum value) of f(x) in [a, b] is obtained at one point in (a, b), it is obvious that this maximum value (or minimum value) is also a maximum value (or minimum value), which is f(x) in (a, b).

(2) the step of finding the maximum and minimum value of f(x) on [a, b]

① Find the extreme value of f(x) in (a, b);

② Compare f(x) with the extreme values of f(a) and f(b), where the largest is the maximum value and the smallest is the minimum value.

5. Life optimization

In life, we often encounter problems such as maximizing profits, saving materials and achieving the highest efficiency. These problems are called optimization problems, and optimization problems are also called maximum problems. Solving these problems has important practical significance. These problems can usually be transformed into function problems in mathematics, and then into the problem of finding the maximum (minimum) value of the function. Such as the derivative of 1. Y = x 2+5x+7 is y.

Make y'>0

Get x & gt-5/2

This means that the function is increasing function 2 in x & gt-5/2. The derivative of y = 4x 2+5x+8 is 8x+5.

Let y'=0, and you get x=-5/8.

The function y is in x; -5/8 is an increasing function.

That is, to the left of x=-5/8, y'

It can be understood that when y' = the root of 0

Check the sign of the root of left and right y'=0. If the left is negative on the right, then f(x) gets the maximum at this root; If the left is negative and the right is positive, then f(x) takes the minimum value at this root.

Now it is to the left of point x=-5/8, Y'

The minimum value obtained here is left negative and right positive.

And f(-5/8)=t (I don't count this number as t)

Then the range of y is [t, +∞)