Find Smallest b to Make Invertible Function: Homework Solution

[Kqwert]

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.

 

[PeroK]

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.

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

 

[Kqwert]

I did find the derivative, i.e

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

 

[PeroK]

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?

 

[Kqwert]

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

 

[PeroK]

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. $b=0, 10, 100$

 

[Kqwert]

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]

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?

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.