Department of Mathematics Maheshtala College Rolles Theorem and

Department of Mathematics Maheshtala College Rolle’s Theorem and the Mean Value Theorem Presented by Pralay Das

Rolle’s Theorem Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b) then there is at least one number c in (a, b) f’(c) = 0. f’(c) means slope of tangent line = 0. Where are the horiz. tangent lines located? f(a) = f(b) a c c b

Ex. Find the two x-intercepts of f(x) = x 2 – 3 x + 2 and show that f’(x) = 0 at some point between the two intercepts. f(x) = x 2 – 3 x + 2 0 = (x – 2)(x – 1) x-int. are 1 and 2 f’(x) = 2 x - 3 0 = 2 x - 3 x = 3/2 Rolles Theorem is satisfied as there is a point at x = 3/2 where f’(x) = 0.

Let f(x) = x 4 – 2 x 2. Find all c in the interval (-2, 2) such that f’(x) = 0. Since f(-2) and f(2) = 8, we can use Rolle’s Theorem. f’(x) = 4 x 3 – 4 x = 0 4 x(x 2 – 1) = 0 x = -1, 0, and 1 Thus, in the interval (-2, 2), the derivative is zero at each of these three x-values. 8

The Mean Value Theorem If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then a number c in (a, b) (b, f(b)) secant line (a, f(a)) a c b represents slope of the secant line.

Given f(x) = 5 – 4/x, find all c in the interval (1, 4) such that the slope of the secant line = the slope of the tangent line. ? But in the interval of (1, 4), only 2 works, so c = 2.