# Given the function #f(x) = x^(1/3) #, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [-5,4] and find the c?

##### 1 Answer

See below.

#### Explanation:

The Mean Value Theorem has two hypotheses:

**H1** :

**H2** :

In this question,

This function is a power function, so it is continuous on its domain, **The function does satisfy the first hypothesis on this interval**

**the function does not satisfy the second hypothesis on this interval.**

(That is, there is at least one point in

Although this function **does not** satisfy the hypotheses on the interval, it does satisfy the conclusion (in two places).

Solving

And after some tedious algebra, we get

# ~~ +- 0.87#

both of which are in

Although this is tedious to work out analytically, it look likely geometrically.

Here is a graph of the function and the secant line

graph{(y-x^(1/3))((y-4^(1/3))/(x-4) - (4^(1/3)+5^(1/3))/9) = 0 [-6.79, 5.7, -4.444, 1.8]}

It certainly looks as if there ought to be two places where the tangent is parallel to the secant. And there are.