Given the function
Let g be the inverse function of f. i.e. g(x)=f−1(x).
what is the entire pblm? In general it is very difficulat to solve a cubic equation see Cubic function - Wikipedia, the free encyclopedia
I would be willing to bet to solve you're given pblm we may not need
to solve for f^(-1)(x) in general
If you want f ^(-1)(5)
f(x) = 5x^3 + 4x -4
note f(1) =5 so f^(-1)(5) = 1
For the derivative [f^(-1)] ' (5) use the theorem on the derivative of the inverse:
if f(a) = c
Then [f^(-1)] ' (c) = 1/f '(a)
In terms of g : g ' (c) =1/f '(a)
In particular [f^(-1)] ' (5) = 1/f '(1) = 1/19