
9-6 Calculus – Differentiation and Integration
STATEMENT If a function f is
a. Continuous in the closed interval [a, b].
b. Derivable in the open interval (a, b).
Then there exists at least one value c Î (a, b) such that
f b f a
b a
f ' c
( ) ( )
( ).
−
−
=
Proof: Now, we define a new function f involving f and designed so as to satisfy the conditions of
Rolle’s theorem.
Consider, f(x) = f(x) + Ax, where a is a constant to be determined such that f(a) = f(b).
∴ + = +
− = −
− =
−
−
f a Aa f b Ab
f b f a A a b
A
f b f a
( ) ( )
( ) ( ) ( )
( ) ( )
.
Now the function f(x) is differentiable, x is differentiable and A being a constant is also differentiable.
Therefore, ...