Find Smallest b to Make Invertible Function: Homework Solution

In summary: 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.
  • #1
Kqwert
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3

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.
 
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  • #2
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.

Why not find the derivative and see what you can do with that?
 
  • #3
I did find the derivative, i.e

f'(x) = 3x^2 + 18x + b. But I am unsure what to do from there.
 
  • #4
Kqwert said:
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?
 
  • #5
It´s positive, but I am a bit confused re. how I should treat b when x is also unknown.
 
  • #6
Kqwert said:
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##
 
  • #7
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##
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?
 
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  • #8
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?

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.
 

1. What is the purpose of finding the smallest b to make an invertible function?

The purpose of this exercise is to determine the minimum value of b that will make the given function invertible. Inverting a function means finding a new function that undoes the original function, resulting in the input and output values being switched. This is an important concept in mathematics and can have various applications in fields such as engineering, physics, and computer science.

2. How do you find the smallest b to make an invertible function?

To find the smallest b, we need to use the concept of the inverse function. The inverse function of a given function is found by switching the input and output variables and solving for the new output variable. In this case, we will be solving for b in terms of the original input and output variables. Once we have the inverse function, we can then find the value of b that makes the function invertible by setting the inverse function equal to the original function and solving for b.

3. What happens if the function is not invertible?

If the function is not invertible, it means that there is no value of b that will make the function invertible. This can happen if the function is not one-to-one, meaning that multiple input values can result in the same output value. In this case, finding the smallest b to make the function invertible is not possible and the function remains non-invertible.

4. Can we use any value of b to make the function invertible?

No, we cannot use any value of b to make the function invertible. The value of b must be carefully chosen to ensure that the inverse function exists. This means that the function must be one-to-one and have a unique inverse function. Otherwise, the function will not be invertible for any value of b.

5. How can finding the smallest b to make the function invertible be useful?

As mentioned earlier, the concept of invertible functions has various applications in different fields. For example, in computer science, invertible functions are used in cryptography to ensure secure data transmission. In mathematics, invertible functions can help simplify complex equations and make them easier to solve. Therefore, finding the smallest b to make a function invertible can be useful in solving real-world problems and applications.

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