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18. 已知函数 $f(x) = \ln \frac{x}{2-x} + ax + b(x-1)^3$
(1) 若 $b=0$，且 $f'(x) \ge 0$，求 $a$ 的最小值；
(2) 证明：曲线 $y=f(x)$ 是中心对称图形；
(3) 若 $f(x) > -2$ 当且仅当 $1 < x < 2$，求 $b$ 的取值范围.
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We need to prove that the curve y equals f of x is centrally symmetric. First, let's determine the domain of the function. For the logarithm ln of x over 2 minus x to be defined, we need x over 2 minus x to be positive. This occurs when x is between 0 and 2. Notice that this domain is symmetric about x equals 1, which suggests the center of symmetry might be at x equals 1.
Now let's establish the condition for central symmetry. A curve y equals f of x is symmetric about a point x naught, y naught if for any h in the domain, f of x naught plus h plus f of x naught minus h equals 2 times y naught. We will test for symmetry about the point 1, a. The symmetric points 1 minus h and 1 plus h should satisfy this condition.
Now let's calculate f of 1 plus h and f of 1 minus h. For f of 1 plus h, we substitute into our function to get ln of 1 plus h over 1 minus h, plus a times 1 plus h, plus b times h cubed. For f of 1 minus h, we get ln of 1 minus h over 1 plus h, plus a times 1 minus h, plus b times negative h cubed. Using the logarithm property, ln of 1 minus h over 1 plus h equals negative ln of 1 plus h over 1 minus h. When we add these expressions, the logarithm terms cancel out, and we get 2a.
Now let's verify our result. Since f of 1 plus h plus f of 1 minus h equals 2a, which is a constant value, the symmetry condition is satisfied. We can calculate f of 1 directly: f of 1 equals ln of 1 over 1, plus a times 1, plus b times 0 cubed, which simplifies to ln of 1 plus a plus 0, equals a. Therefore, the center of symmetry is at the point 1, a. This proves that the curve y equals f of x is indeed centrally symmetric about the point 1, a.
Our proof is now complete. We have successfully demonstrated that the curve y equals f of x is centrally symmetric about the point 1, a. The key insight was recognizing that the domain 0 to 2 is symmetric about x equals 1, then applying the central symmetry condition. By calculating f of 1 plus h plus f of 1 minus h, we found it equals the constant 2a, which satisfies the symmetry requirement. Finally, we verified that f of 1 equals a, confirming that the center of symmetry is indeed at the point 1, a. This completes our proof.