Let f be the decreasing function defined by f(x)=−x3−6x2−12x+8, where f(4)=−8. If g is the inverse function of f, which of the following is a correct expression for g′(−8) ?

Answer 1

Sorry, f(4) does NOT equal -8, it equals -200. You could see this easily on a graphing calculator, or open a spreadsheet and list f(x) from say x = -20 to 20. It's not a trivial problem, since it appears that f(x) may NOT be monotonically decreasing; in particular, the region from x = -3 to -1 appears suspect!

You could easily check this by taking f '(x), factors to -3 * (x^2-4x-4) which has roots at -6 (1 + 2^0.5), hence expect at least a small zone of positive slope on the original f function, hence a general form of an inverse function does not exist!

-- Cheers, --Mr. d.

Answer 2

If f(4) = - 8 then inverse function g(- 8) = 4. So, g'(- 8) = 1/f'(g(- 8)) = 1/f'(4) ; f'(x) = - 3x² - 12x - 12; f'(4) =

- 3·4² - 12·4 - 12 = - 108. So g'(- 8) = - 1/108.