(C) Suppose that f(x) is continuous on [a, b], differentiable on (a, b), and define g (x ) = f (2 ) - f ( a ) + f ( b ) - f ( a ) b - a " (x - a ) Use Rolle's Theorem to show that g'(c) = 0 for some c in (a, b). (D) Use the previous result to show that there is c in (a, b) satisfying f (b ) - f(a ) .= f'(c) b - a You have now proved the Mean Value Theorem. (E) Define the function f (2) = -xIn(x) if x > 0 if x = 0 on the interval [0, co), let c be defined by the equation f (b) - f(0) = f'(c) b - 0 for b > 0, and complete the following table accurate to five decimal places: b 2 1 0.1 0.01 0.001 What do you expect lim $ to equal? 6-+0+