证明:
1) 由积分中值定理,有
2 f(0) = (0,2) ∫ f(x)dx = (2-0) * f(c1) , ( 0 < c1 < 2 )
故 f(c1) = f(0)
2) 由原式有
[ f(2) + f(3) ] / 2 = f(0)
由连续性,知在 (2, 3) 之间存在一点 x = c2, 使得 ,
f(c2) = f(0)
3) 由罗尔定理,得,
f '(η1) = 0 , ( 0 < η1 < c1 )
f '(η2) = 0 , ( c1 < η1 < c2 )
4) 由罗尔定理,得,
f " (ξ) = 0 , ( η1 < ξ < η2 )