How do you determine whether the function satisfies the hypotheses of the Mean Value Theorem for #f(x)=x^(1/3)# on the interval [-5,4]?
1 Answer
It doesn't. But it does satisfy the conclusion. See below.
Explanation:
There are two hypotheses for MVT.
The function must be continuous on
The function must be differentiable on
The first is true (satisfied) because
This function fails to be differentiable at
This function does not satisfy the hypotheses of the Mean Value Theorem on this interval.
Bonus material
This function DOES satisfy the conclusion of the MVT on this interval. We cannot use the Mean Value Theorem to conclude that there is a
We can, however solve
The algebra is be tedious to type, but the graph makes this plausible.
It shows
graph{(y-x^(1/3)) (y-4^{1/3)-((4^(1/3)+5^(1/3))/9)(x-4))= 0 [-6.06, 5.04, -2.864, 2.685]}