1, Rolle mean value theorem: If f(x) satisfies: (1) is continuous on [a, b]; (2) Derivable on (a, b); (3)f(a)=f(b)。 Then at least c∈(a, b) exists, so f(c)'=0.

2. Lagrange mean value theorem: If f(x) satisfies: (1) is continuous on [a, b]; (2) Derivable in (a, b). Then at least c∈(a, b) exists so that f(b)-f(a)=f'(c)(b-a) or f(a+h)-f(a)=f'(a+θh), where h=b-a, 0.

3. Cauchy mean value theorem: If f(x) and g(x) satisfy: (1) is continuous on [a, b]; (2) Derivable in (a, b); (3)g'(x)≠0。 Then at least c∈(a, b) exists, so that [f (b)-f (a)]/[g (b)-g (a)] = f' (c)/g' (c).