Question

To illustrate the Mean Value Theorem with a specific function, let's consider f(x) = x^3 − x, a = 0, b = 5. Since f is a polynomial, it is continuous and differentiable for all x, so it is certainly continuous on [0, 5] and differentiable on (0, 5). Therefore, by the Mean Value Theorem, there is a number c in (0, 5) such that

f(5) − f(0) = f '(c)(5 − 0).

Now f(5) = ______ , f(0) = ______, and f '(x) = ______ , so this equation becomes

______ = f '(c)(5) =(_____) 5 = ______ ,

which gives c^2 = ______ , that is, c = ± _______ . But c must be in (0, 5), so c = ______ .

The figure illustrates this calculation: The tangent line at this value of c is parallel to the secant line.

Answer 1