Determine if the Mean Value Theorem for Integrals applies to the function

Question

f(x) = x^3 − 9x on the interval [−1, 1]. If so, find the x-coordinates of the point(s) guaranteed to exist by the theorem

Answer 1

Step 1 :

The function is .

The interval is .

Mean value theorem for integrals :

is continuous on closed interval then there exists a number  in the closed interval such that .

As the function satisfies the mean value theorem , .

Apply fundamental theorem :.

.

Step 2 :

Find the value of .

The value of exists between , therefore  is not to be considered.

Hence is the solution in the interval .

The value of represents the -coordinate in the function.

.

The -coordinate of the point satisfying mean value theorem is .

Solution :

The -coordinate of the point satisfying mean value theorem is .