# Given the function #f(x)=(x-3)^(2/3)#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [1,4] and find the c?

##### 1 Answer

Please see below.

#### Explanation:

The Mean Value Theorem has two hypotheses:

**H1** :

**H2** :

We say that we can apply the Mean Value Theorem if both hypotheses are true.

**H1** : The function

Because power functions are continuous on their domains and linear functions are continuous everywhere. And the composition of continuous functions is continuous.

**H2** : The function

Because the derivative,

**A note on "if . . . then . . . " theorems**

If the hypotheses are not true, we learn *nothing* about the truth of the conclusion.

To determine whether the conclusion is true or false we try to solve the equation in the conclusion of MVT.

Solving

Here is a graph of the function and the secant line through the points with

graph{(y-(x-3)^(2/3))((y-1)-(1-4^(1/3))/(3)(x-4))=0 [-15.1, 10.21, -5.03, 7.63]}