# Rolle's Theorem

How to solve the Rolle's theorem problems: theorem, example, and its solution.

## Theorem

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

differentiable in (*a*, *b*), and *f*(*a*) = *f*(*b*),

then *c* exists in (*a*, *b*)

that satisfies *f*'(*c*) = 0.

In other words,

if the graph satisfies these three hypotheses,

no matter how the graph looks like,

there's always a point whose tangent's slope is 0.

## Example

State the three hypotheses of the Rolle's theorem:

1. *f*(*x*) is coninuous in [0, *π*].

2. *f*(*x*) is differentiable in (0, *π*).

3. *f*(0) = *f*(*π*)

Then, by the Rolle's theorem,*c* exist in (0, *π*)

that satisfies *f*'(*c*) = 0.

To find the *c*,

draw *y* = sin *x*, [0, *π*].

Then, you'll see that

at *c* = *π*/2,

the slope of the tangent line, *f*'(*π*/2), is 0.

Or, to find *c*,

you can solve [sin *c*]' = 0,

which is cos *c* = 0.

Then *c* = *π*/2.

Derivative of sin *x*

Solving trigonometric equations