Chapter 4.2, Problem 57E

Generalized Mean Value Theorem Suppose the functions f and g are continuous on ⌈a, b⌉ and differentiable on (a, b), where g(a) ≠ g(b). Then there is a point c in (a, b) at which

$\frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f\prime (c)}{g\prime (c)}.$

This result is known as the Generalized (or Cauchy’s) Mean Value Theorem.

1. a. If g(x) = x, then show that the Generalized Mean Value Theorem reduces to the Mean Value Theorem.
2. b. Suppose f(x) = x² − l, g(x) = 4x + 2, and [a, b] = [0, 1]. Find a value of c satisfying the Generalized Mean Value Theorem.