Question

Verify that f has an inverse. Then use the function f and the given real number a to find (f^-1) ' (a).

f (x) = x^3 - 1,     a = 26

Answer 1

Step 1:

The function is .

The function is continuous in the interval .

Derivative of is , which is positive in the interval .

So the function is one to one function and is strictly monotonic.

Find the inverse of the function.

Consider , then .

Substitute in the above expression.

.

Interchange .

Then .

Inverse of the function is .

Step 2:

The inverse function is

Apply derivative on each side with respect to .

.

Find .

.

Solution :

The function is monotonic and has an inverse .

.