Answer:
There is a value of
in (-1, 1),
.
Explanation:
Let
for
, we need to prove that
is continuous and differentiable to apply the Mean Value Theorem. Given that
is a polynomical function, its domain comprises all real numbers and therefore, function is continuous.
If
is differentiable, then
exists for all value of
. By definition of derivative, we obtain the following expression:





(Eq. 2)
The derivative of a cubic function is quadratic function, which is also a polynomic function. Hence, the function is differentiable at the given interval.
According to the Mean Value Theorem, the following relationship is fulfilled:
(Eq. 3)
If we know that
,
and
, then we expand the definition as follows:




There is a value of
in the interval (-1, 1),
.