## Mean Value Theorem

For function f differentiable in the open interval (a, b) and continuous on the closed interval [a, b], there exists a point c between a and b that satisifies:

f (b) − f (a) = f '(c)(b − a)

Example:

Example:

For f (x) = x

^{2}in the interval [1, 7], determine the value of c, which satisifies the Mean Value Theorem.Solution:

^{}f (x) = x

f '(x) = 2x

a = 1

b = 7

f (a) = f (1) =1

f (b) = f (7) = 7

Using the Mean Value Theorem:

49 − 1 = 2c (7 − 1)

48 = 2c (6)

c = 4

^{2}f '(x) = 2x

a = 1

b = 7

f (a) = f (1) =1

^{2}= 1f (b) = f (7) = 7

^{2}= 49Using the Mean Value Theorem:

49 − 1 = 2c (7 − 1)

48 = 2c (6)

c = 4