【正确答案】必要性：设(*)成立，x₁，x₂∈(a，b)且x₁＜x₂
f(x₂)＜f(x₁)+f'(x₁)(x₂-x₁)，f(x₁)＜f(x₂)+f'(x₂)(x₁-x₂)．
两式相加 [f'(x₁)-f'(x₂)](x₂-x₁)＞0
f'(x₁)＞f'(x₂)，即f'(x)在(a，b)单调减少．
充分性：设f'(x)在(a，b)单调减少．对于