# Invertible function

## Homework Statement

Find the smallest value of b so that the function f(x) = x^3 + 9x^2 + bx + 8 is invertible.

## 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.

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PeroK
Homework Helper
Gold Member

## Homework Statement

Find the smallest value of b so that the function f(x) = x^3 + 9x^2 + bx + 8 is invertible.

## 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.
Why not find the derivative and see what you can do with that?

I did find the derivative, i.e

f'(x) = 3x^2 + 18x + b. But I am unsure what to do from there.

PeroK
Homework Helper
Gold Member
I did find the derivative, i.e

f'(x) = 3x^2 + 18x + b. But I am unsure what to do from there.
What do you know about an increasing function and its derivative?

It´s positive, but I am a bit confused re. how I should treat b when x is also unknown.

PeroK
Homework Helper
Gold Member
It´s positive, but I am a bit confused re. how I should treat b when x is also unknown.
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. $b=0, 10, 100$

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. $b=0, 10, 100$
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
PeroK
Yes, any less than 27 and the derivative is negative on an interval. You might like to think about the invertibility of a function that has an inflection point, as is the case for $b =27$ here.