Question

Let f(x) = x^3 + 2x^2 + 5x + 1, and let g be the inverse function of f. What is the value of g '(9)?

Answer 1

I think I responded on one of your questions.

Answer 2

I don't really want to type the same thing I did there here.

Answer 3

http://openstudy.com/study#/updates/54596ee0e4b04ccc259206a7

Answer 4

\[
\text{If g(x) is the inverse fnction of f(x), then:}\\
g'(x)=\frac{1}{f'(g(x))} \\
g'(9)=\frac{1}{f'(g(9))} ~~~\text{------ (1)}\\
f(x) = x^3 + 2x^2 + 5x + 1 \\
\text{Normally we are given x, and we find y. }\\
\text{In inverse function we are given y and we need to find x.} \\
\text{To find g(9), set y = 9 and find x.} \\
9 = x^3 + 2x^2 + 5x + 1 \\
x^3 + 2x^2 + 5x - 8 = 0 \\
\text{Rational roots theorem: Factors of 8 are: }\pm(1, 2, 4, 8). \\
\text{Plug each factor into }x^3 + 2x^2 + 5x - 8 = 0 \text{ and see which one}\\
\text{makes it zero. That x value will be g(9).} \\
\text{Then find f '(x) and plug it into equation (1) } \\
\]