The problem of finding the range of parameters often appears in the finale of derivative in college entrance examination. We all know that its general solution is classified discussion, which is the most basic method, but its calculation process is often very complicated. Separating parameters, constructing new functions and using the boundedness of function images are also common solutions, but the method is not easy, not to mention that not all topics can separate parameters.
However, if the function value at the end of the interval satisfies certain particularity (usually the function value at the end is 0), we can use another simple and effective solution: the end effect to solve the range of parameters. The problem-solving process can generally be divided into two steps: consider whether the final value of the function has particularity in the interval and narrow the scope; Find the range of parameters through the necessary conditions of inequality.
By solving the monotonicity of the judgment function, it is proved that the necessary condition is the sufficient condition: if the inequality f(x, m)≥0 and m is a parameter, it is a constant in the interval [a, b], and the value range of m is found. The first step is to narrow the range of values: if the function value at the end of the interval is not 0, that is, f(a)≠0 or f(b)≠0, the end 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;
The function value of the interval endpoint value is of 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) is found. Both the function value and derivative value of the interval endpoint value are of type 0: that is, if f(a)=0 (or f(b)=0) and f'(a)=0 (or f'(b)=0), then find the solution f'(a)≥0(f'(b)≥0) of m and the value set d. Use the parameter range m∈D obtained in the first step to find f ′ (x) and f″(x) to judge monotonicity.