Rolle's and Mean value theorem of a function

A function is a special relationship between the inputs (i.e., the domain) and their outputs (known as the codomain), where each input has exactly one output that can be traced back to its input. A function is generally denoted by f(x), where x is the input. A derivative tells us the rate of change with respect to a certain variable. It is primarily used when there is some varying quantity and the rate of change is not constant. The derivative is used to measure the sensitivity of one variable (dependent variable) with respect to another variable (independent variable). Rolle's theorem:- If i) a real-valued function f is continuous on a closed interval [a, b], ii) differentiable on the open interval (a, b), and iii) f (a) = f (b), then there exists “c” in the open interval (a, b) such that f ′(c) = 0. Mean Value Theorem Statement Suppose f(x) is a function that satisfies the following: i) f(x) is continuous in [a,b] and ii) f(x) is differentiable in (a,b) Then, there exists a number c, within a and b and [f(b) -f(a)]/ (b-a) = f ' (c) Special Case: When f(a) = f(b), then at least one c within a, b exists such that f'(c) = 0. This case is known as Rolle’s Theorem.