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Answer 1

Answer:

1. (i) Rolle’s theorem states that if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) such that f(a) = f(b), then f′(x) = 0 for some x with a ≤ x ≤ b. In other words, if a continuous curve passes through the same y-value (such as the x-axis) twice and has a unique tangent line (derivative) at every point of the interval, then somewhere between the endpoints it has a tangent parallel to the x-axis. The theorem was proved in 1691 by the French mathematician Michel Rolle, though it was stated without a modern formal proof in the 12th century by the Indian mathematician Bhaskara II.

(ii) The Mean Value Theorem establishes a relationship between the slope of a tangent line to a curve and the secant line through points on a curve at the endpoints of an interval. It states that If a function f(x) is continuous on a closed interval [a,b] and differentiable on an open interval (a,b), then at least one number c ∈ (a,b) exists such that  f'(c) = $\frac{f(b) - f(a)}{b - a}$