The discussion revolves around determining the smallest value of b for the function f(x) = x^3 + 9x^2 + bx + 8 to be invertible, focusing on the conditions for the function to be strictly increasing or decreasing.
Some participants have suggested graphing the derivative for various values of b to understand its behavior. There is an exploration of the relationship between the derivative and the conditions for invertibility, with some participants noting specific values of b and their effects on the derivative.
Participants are considering the implications of the derivative being zero at certain points and how that relates to the function's invertibility, particularly in the context of inflection points.
Kqwert said:Homework Statement
Find the smallest value of b so that the function f(x) = x^3 + 9x^2 + bx + 8 is invertible.
Homework Equations
The Attempt at a Solution
I know that the function has to be only increasing/decreasing, and I think it is needed to find the derivative of the function. I do however not know how b should be found.
Kqwert said:I did find the derivative, i.e
f'(x) = 3x^2 + 18x + b. But I am unsure what to do from there.
Kqwert said:It´s positive, but I am a bit confused re. how I should treat b when x is also unknown.
Thanks. For b = 27 the derivative is zero at x = -3, but is positive for all other values of x. Is this the correct answer?PeroK said:Okay, so you need ##f'(x)## to be generally positive. Can you graph the function ##f'(x)## for some values of ##b## to see what's happening? E.g. ##b=0, 10, 100##
Kqwert said:Thanks. For b = 27 the derivative is zero at x = -3, but is positive for all other values of x. Is this the correct answer?