Secant line, continuous function, differentiable functions, Rolle's theorem, Mean Value TheoremTheorem: Let f(x) be a function continuous on [a,b] and differentiable on (a,b). Then there is a c in (a,b) such that f'(c) = (f(c)-f(a))/(b-c) Proof: the theorem follows by applying Rolle's theorem to the auxiliary function h(x) = -(x-b)(f(x) – f(a)) Here is a geometric interpretation: The triangle formed by the x axis, the tangent line through (c, f(c)), and the secant line through (c, f(c)) and the point (b, f(a)) is an isosceles triangle (the green triangle). Therefore the slopes of the two sides not on the x axis are f'(c) and –f'(c) The example used is the function f(x) = c0 + c1 x + c2 x^2 + c3 x^3Componente Curricular::Educação Superior::Ciências Exatas e da Terra::Matemática