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

1 Answer

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.