Verify that f has an inverse. Then use the function f and the given real number a to find (f −1)'(a). (Hint: See Example 1. If an answer does not exist, enter DNE.) Function Real Number f(x) = 1 6 (x5 + 8x3) a = 16

In this exercise we have to use the knowledge of inverse function to calculate the equations and functions given, in this way we find that:

A)

B)

Given the function:

We need to find:

A function has its inverse function existing lf and only if the given function

fis one-to-one fuction: The given function:

Derivate the equation we have that:

Since and are always positive, so the function is always monetonic

increasing  and it passed the horizontal line test. Therefore, the function is one to one. Then, since the function, is one-to-one, It has inverse. Using the formula:

Since we want to find :

Let take the inverse of both sides:

Because when we set a function into as inverse function, it gives our original input value as:

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