Rolle’S Theorem: How It Relates The Behavior Of A Differentiable Function To Its Roots.

Rolle’s Theorem

1) f must be continuous on[a,b] and differentiable on (a,b). 2) f(a)=f(b) then there is at least one number c in (a,b) such that f'(c)=0

Rolle’s Theorem is a fundamental theorem of calculus introduced by the French mathematician Michel Rolle in the 17th century. It is a theorem that relates the behaviour of a differentiable function to its roots or zeroes.

Statement of Rolle’s Theorem:
Let f(x) be a continuous function on the closed interval [a,b] with f(a) = f(b), then there exists at least one c, a < c < b, such that f'(c) = 0. Explanation: The above statement of Rolle's Theorem implies that if a function f(x) is continuous on the interval [a,b] and has equal values at a and b, then it must have at least one critical point or stationary point between a and b where its derivative, f'(x), is zero. In other words, there must be one or more points in the interval where the function's slope is flat (horizontal line) or the tangent to the curve is parallel to the x-axis. This theorem is useful in many fields, such as optimization and numerical analysis, to find the solutions of equations, calculate extreme points, and construct graphs with higher precision. Example: Let f(x) = x^3 - 3x^2 + 6 defined over the interval [1,3]. It is a polynomial function that is continuous and differentiable everywhere. We can use Rolle's theorem to show that there exists at least one critical point or a root of f'(x), where the tangent is parallel to the x-axis. We have f(1) = 4 and f(3) = 0, so f(1) = f(3). Therefore, we can apply Rolle's Theorem to conclude that there is at least one value of c in the interval (1,3) where f'(c) = 0. Differentiating f(x), we get: f'(x) = 3x^2 - 6x Setting f'(c) = 0, we get: 3c^2 - 6c = 0 Solving for c, we get: c = 0 or c = 2 However, c = 0 is not in the interval (1,3). Therefore, the only critical point is at c = 2, where f'(2) = 0. This is where the tangent line is horizontal and intersects the graph of f(x) at a local minimum.

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