Rolle's Theorem

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.

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

For the given f(x), show that c exists in (0, pi) that satisfies f'(c) = 0. f(x) = sin x

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