I. Proof of definition

Steps to prove monotonicity of function by definition:

① arbitrary value: let x 1 and x2 be any two values in this interval, X 1

② Difference deformation: make the difference f(x2)-f(x 1), and transform the difference into a symbol that is favorable for judging the difference by factorization, formula and physical and chemical methods.

③ Judgment and numbering: determine the symbol of f(x2)-f(x 1).

④ Draw a conclusion: draw a conclusion according to the definition (if the difference is greater than; 0, which is an increasing function; If the difference is

That is, "take any value-do differential deformation-judge the number-draw a conclusion"

Second: Composite function

1. The sum of the two increasing function is still increasing function;

2. The decreasing function of increasing function is increasing function;

3. The sum of two subtraction functions is still a subtraction function;

4. The subtraction function minus increasing function is a subtraction function;

And:

When the function values have the same sign in the interval, the reciprocal of the increase (decrease) function is the decrease (increase) function.

Three: Derivation

Any two points in the interval a

f(b)-f(a)=f'(c)(b-a).

Now f'(c)>0, b-a >: 0, therefore

F(b) > female (a).

Since a and b are arbitrary, by definition, f(x) increases in this interval.

When f' (x) < 0, we can see from the above proof process that f(x) is decreasing at this time.