1. The original function f(x) is monotonically decreasing, because when x gradually approaches b, the function value f(x) tends to 0, and f(x) is a function with continuous first derivative. According to the property that the derivative of monotone increasing function is greater than or equal to 0, we can draw a conclusion.
2. The original function f(x) satisfies f (x)+f (x) 2- 1
Now let's prove that b-a > 2, that is, the interval length is greater than 2.
Reduction to absurdity: suppose B-A
According to the monotone decreasing property, for any x ∈ (a, b), there exists f(x_n) ≥ f(x), so lim f(x_n) ≥ lim f(x).
However, according to the topic conditions, lim f(x_n) should be equal to 0, which is inconsistent with lim f(x), so the assumption is not established, that is, there is B-A >; 2。
Next, let's give an example to prove that this equation holds:
When a = -∞, b = +∞, f (x) = (1-e x)/(1+e x), the subject condition is satisfied. Because:
-lim f(x)= lim(( 1-e^x)/( 1+e^x))= lim((-e^x)/2)=-∞，