1549901369-Elements_of_Real_Analysis__Denlinger_

6.4 Mean-Value Type Theorems 323

Theorem 6.4.3 (Mean Value Theorem, "MVT") Suppose f: [a, b] -t JR

is differentiable on (a,b), and continuous on [a,b], where a< b. Then :Jc E

(a, b) 3 f'(c) = f(b~ =~(a).

y

f(b)-j(a)

(b,f(b)) '} y=f(x)+ b-a (x-a)

____ J f(b)-f(a)

a c c

Figure 6.7

I I

I I

b x

Proof. Suppose f is differentiable on (a, b), and continuous on [a, b], where

a< b. Define a new function hon [a, b] by

h(x) = f(x) - f(a) - f(b~ =~(a) (x - a). (3)

Then h differs from f by a first degree polynomial function in x, so h is

continuous wherever f is and differentiable wherever f is. Moreover,

f(b) - f(a)

h(a) = f(a) - f(a) - b _a (a - a) = 0, and

h(b) = f(b) - f(a) - f(b) - f(a) (b - a)

b-a

= f(b) - f(a) - [f(b) - f(a)]

= 0.

Thus, h(a) = h(b), and h satisfies all the hypotheses of Rolle's theorem.
Hence, by Rolle's theorem, 3 c E (a, b) 3 h'(c) = 0. Now, from (3),

h'(x) = f'(x) _ f(b) - f(a).

b-a