1. Rolle’s Theorem
By Tae In Ha

2. What is Rolle’s Theorem?
A theorem made by a French mathematician Michel Rolle (Hence his last name) during The theorem states:
Let f be continuous on the closed interval
[a, b] and differentiable on the open interval
(a, b). If
f(a) = 0 and f(b) = 0
Then there is at least one point c in the interval
(a, b) such that f’(c) = 0.

3. In other words…
Rolle’s theorem only applies if the graph of a differentiable and continuous function intersects the x-axis at points a and b. In the interval between a and b, a horizontal tangent line must exist.

4. An example Let’s say f(x) = X2-2X-3; [-1,3]
Using the Rolle’s theorem, plug -1 and 3 into the original function to
verify if it is equal to 0.
f(-1) = (-1)2-2(-1)-3 = 0
f(3) = (3)2-2(3)-3 = 0
Now that both are equal to 0, derive f(x).
f’(x) = 2X-2 <= Set it equal to zero to find the horizontal tangent line.
2X-2 = 0 X=1
Therefore, C = (-1,3)

5. Now you do it! Solve: f(x) = x2 – 4x + 3; [1,3]
f(x) = cosx; [π/2, 3π/2]

6. ANSWERS f(1) = 0, f(3) = 0; f’(x) = 2x – 4 2x – 4 = 0; x = 2
Therefore, c = 2 in (1,3)
f(2) = -12
Therefore, Rolle’s theorem does not apply.
f(π/2) = 0, f(3π/2) = 0; f’(x) = -sinx
-sinx = 0; x = π in (π/2, 3π/2)

7. Sources
CALCULUS (8th Ed.) by Howard Anton