# Rolleâ€™s theorem

If a function*f*(

*x*) is

- continuous in the closed interval [
*a*,*b*], i.e., continuous at each point in the interval [*a*,*b*] - differentiable in an open interval (
*a*,*b*), i.e., differentiable at each point in the open interval (*a*,*b*) *f*(*a*) =*f*(b)

then there will be at least one point

*c*, in the interval (*a*,*b*) such that*f*â€²(*c*) = 0.â€‹

**Converse of Rolleâ€™s theorem is not true, i.e., if a function**

*Note:**f*(

*x*) is such that

*f*â€²(

*c*) = 0 for at least one

*c*in the open interval (

*a*,

*b*) then it is not necessary that

*f*(*x*) is continuous in [*a*,*b*]*f*(*x*) is differentiable in (*a*,*b*)*f*(*a*) =*f*(*b*)

For example, we consider the function

*f*(*x*) =*x*^{3}â€“*x*^{2}â€“*x*+ 1 and the interval [â€“1, 2].Here,

*f*â€²(*x*) = 3*x*^{2}â€“ 2*x*â€“ 1 âˆ´*f*â€²(1) = 3 â€“ 2 â€“ 1 = 0 and I âˆˆ (â€“1, 2)But condition (3) of Rolleâ€™s theorem is not satisfied since

*f*(â€“1) â‰*f*(2).

# Lagrangeâ€™s mean value theorem

If a function*f*(

*x*) is

- continuous in the closed interval [
*a*,*b*], i.e., continuous at each point in the interval [*a*,*b*] - differentiable in the open interval (
*a*,*b*), i.e., differentiable at each point in the interval (*a*,*b*)

then there will be at least one point

*c*, where*a*<*c*<*b*such that*f*â€²(

*c*) =