If f(x) = |(x2 − 9)(x2 + 1)|, how many numbers in the interval [−1, 1] satisfy the conclusion of the mean value theorem?

only one number c=0 in the interval [-1,1]

Step-by-step explanation:

Given : Function   in the interval [-1,1]

To find : How many numbers in the interval [−1, 1] satisfy the conclusion of the mean value theorem.

Mean value theorem : If f is a continuous function on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point 'c' in (a,b) such that

Solution : f(x) is a function that satisfies all of the following :

1) f(x)  is continuous on the closed interval [-1,1]  

2) f(x) is differentiable on the open interval  (-1,1)

Then there is a number  c such that  

........[1]

Now, we find f'(x)

Differentiating w.r.t  x

In place of x we put x=c

 (by [1], f'(c)=0)

either c=0 or  

we cannot take  because they don't lie in the interval [-1,1]

Therefore, there is only one number c=0 which lie in interval [-1,1] and satisfying the conclusion of the mean value theorem.