Rolle’s Theorem By Tae In Ha

What is Rolle’s Theorem? A theorem made by a French mathematician Michel Rolle (Hence his last name) during 1652-1719. 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.

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.

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 (-1,3)

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

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)

Sources CALCULUS (8th Ed.) by Howard Anton