Question

How do you verify that the hypothesis of the Mean Value Theorem are satisfied for `f(x)=sqrt(25-x^2)`?

Answer 1

Refer to explanation

Explanation:

The hypothesis of the Mean Value Theorem requires that the function be continuous on some closed interval [a, b] and differentiable on the open interval (a, b).

The domain of the function is for all x reals that

`25-x^2>=0=>D(f)=[-5,5]`

Computing the derivative we get that

`f'(x)=-x/(sqrt(25-x^2))`

we see that is differentiable on the open `(-5,5)`

Hence MVT is satisfied.