Answer: The only solution to this is x = 0. Therefore, g(1) = 0 and g' (1) = g' (1) = 1/f ' (0) = 1/2.
Step-by-step explanation: Since, f(x) = 3x3 + 2x + 1, it follows that f ' (x) = 9x2 + 2. So by the above theorem,
g' (1) = 1/f ' (g(1)) where f ' (g(1)) = 9 · g(1)2 + 2.
It remains to figure out what g(1) equals. Since g(x) = f -1 (x), g(1) is the solution to
3x3 + 2x + 1 = 1 <----> (3x2 + 2)x = 0