How Rolle’S Theorem Helps Find Horizontal Tangents And Multiple Roots Of Polynomial Equations In Calculus

Rolle’s Theorem

if f(x) is continuous on [a,b] and differentiable on (a,b), and if f(a)=f(b), then there is at least one point (x=c) on (a,b) where f'(c)=0

Rolle’s Theorem is a fundamental concept in calculus that is typically used for finding a point within an interval where a function has a horizontal tangent line. More formally, the theorem states that if f is a continuous function on the closed interval [a,b] and differentiable on the open interval (a,b), and if f(a) = f(b), then there exists at least one point c in (a,b) where f'(c) = 0. In other words, if a function has equal values at the endpoints of an interval, then somewhere within that interval, the function must have a horizontal tangent line.

To understand the theorem more deeply, it is important to note that if a function has a horizontal tangent line at a particular point c, then f'(c) = 0. This is because the derivative of a function at a specific point is the slope of the tangent line at that point, and if the tangent line is horizontal, its slope is zero. Conversely, if f'(c) = 0, then the tangent line at c is horizontal, so the derivative being zero at a point guarantees the existence of a horizontal tangent line at that point.

One of the simpler applications of Rolle’s Theorem is to prove the existence of multiple roots of a polynomial equation within an interval. For example, if p(x) = x^2 – 2x + 1, then the function is continuous and differentiable everywhere, and p(0) = 1 = p(1). By Rolle’s Theorem, there must exist at least one value c in (0,1) such that p'(c) = 0. Indeed, p'(x) = 2x – 2, so p'(c) = 0 when x = 1. Therefore, the theorem guarantees that there exists at least one value of x between 0 and 1 where p(x) = 0.

Overall, Rolle’s Theorem is a crucial tool for finding points of interest in the behavior of functions, especially in understanding the root of polynomial equations and the behavior of curves in general.

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