How to use the derivative to see if the function is invertible and how to find the inverse of the derivative

Question

So the function I was given is f(x)= 2x^5 +3x^3+x

 

it is pretty easy to find the derivative using the power rule however on part b of the question it asks how can you use your answer (f'(x)=10x^4 +9x^2 +1) to determine if f(x) is invertible. I don't know how to rewrite this to see if it could be.

 

part e) find (f^-1)'(6)

 

so I know this is where the dependent and independent variables switch but I have no idea how to do that for the derivative.

 

If you could help me on these parts I would be very grateful. Thank you.

Answer 1

A function is invertible if it is one-to-one. A strictly increasing function, or a strictly decreasing function, is one-to-one. 

 

If you can demonstrate that the derivative is always positive, or always negative, as it is in your problem, then you've shown that the function is one-to-one, hence invertible. Technically, the derivative can also be zero at isolated points, just not zero on an interval.

 

For your case, note that the derivative is a sum of squares and a positive constant, hence is, always positive.

Answer 2

Theorem on Derivative of Inverse of a Function:
For monotonic continuous function with an inverse, the following is true:
If c is a value of x in the interval [a, b] and f(c) = d, then (f^-1)'(d) = 1/f'(c).

 

One can see by inspection that f(1) = 6.

Therefore, using the above theorem, with c=1, d=6,

we find that the derivative of the inverse of f, evaluated at 6, is the reciprocal of f'(1); in this

case that gives the answer 1/20.