The process of solving problems can generally be divided into two steps:

(1) Consider whether the endpoint value of the function has particularity in the interval, and narrow the range: find the value range of the parameter through the necessary conditions for the inequality to be established.

(2) By solving the monotonicity of the judgment function, it is proved that the necessary condition is a sufficient condition.

Let me describe the problem-solving method in detail:

Topic: If the inequality f(x, m)≥0, m is a parameter and constant in the interval [a, b], find the range of m.

Step 1, narrow the range of values:

Divided into three situations:

If the function value at the endpoint of (1) interval is not 0, that is, f(a)≠0 or f(b)≠0, the endpoint effect cannot be used. However, since the inequality f(x, m)≥0 always holds in the interval [a, b], it also holds at the endpoint, that is, the parameter range can be narrowed by applying f(a)≥0 and f(b)≥0;

(2) The interval endpoint function value is type 0: if f(a)=0 (or f(b)=0), but f'(a)≠0 (or f'(b)≠0), then the solution f'(a)≥0(f'(b)≤0.

(3) The function value and derivative value of the interval endpoint value are both of type 0: that is, if f(a)=0 (or f(b)=0) and f'(a)=0 (or f'(b)=0), the solution f'(a)≥0(f'(b)≥0) is obtained.

The second step is to prove sufficiency:

Using the parameter value range m∈D obtained in the first step, find f ′ (x) and f″(x) to judge monotonicity, and then judge whether the inequality f(x, m)≥0 is constant.

Second, the detailed explanation of endpoint effect examples

1, and the interval endpoint value is zero:

2. The function value and derivative value of the interval endpoint value are of type 0: