If f(a) = f(b) then,
There exists a real number c ∈ (a, b) such that f'(c) = 0, This special case called as Rolle’s theorem.
Rolle’s theorem essentially states that any real-valued differential function that attains equal values at two distinct points on it, must have at least one stationary point somewhere in between them, that is a point where the first derivative (the slope of the tangent line to the graph of a function) is zero.
Let f(x) is a real-valued function defined on [a, b] and it is continuous on [a, b]. This means that we can draw the graph of f(x) between the value of x = a and x = b also f(x) is differentiable on (a, b) which means the graph of f(x) has a tangent at each point of (a, b). Now the existence of real number c ∈ (a. b) such that f'(c) = 0 shows that the tangent to the curve at x = c has slope 0, that is tangent is parallel to x-axis since f(a) = f(b).
