Finding Extrema of a Set: A Derivative Approach

[JulienB]

Homework Statement

Hi everybody! It's kind of the first time that I try to solve this type of problem, so I'd like to see what you guys think about it and if I am heading in the right direction:

Examine if the set {(x² + x + 1)/(x² + 1) : x∈ℝ} has a maximum and/or a minimum, and calculate those extrema if so.

Homework Equations

Derivatives?

The Attempt at a Solution

Okay so I directly have a first question regarding the problem: can I treat the set as a function or is that nonsense? I think the derivatives of that expression should tell me where the slope of the function is zero and therefore potentially reaching an extremum (or not), right?

So here we go:

(d/dx)(x² + x + 1)/(x² + 1) = [(2x + 1)(x² + 1) - (x² + x + 1)(2x)]/(x² + 1)² = (-x² + 1)/(x² + 1)²

which is equal to 0 for x = {-1;1}. Now I take the second derivative of the set to check if it is also equal to 0 for x = {-1;1}, and if it is not then those values of x would mark two extrema.

(d/dx)(-x² + 1)/(x² + 1)² = [2x(x² - 3)]/(x² + 1)³

which is equal to 1/2 for x = -1 and -1/2 for x = 1, which would that the set has a minimum at x = 1 and a maximum at x = -1. But I know from the graph that that is wrong, although the values for x are right :/

Where is my mistake? Is that to begin with the right way to proceed? And if yes, are there also other methods? Do you have any remark about such problems, especially about the difference between extrema of sets and of functions?Thank you in advance for your answers.Julien.

 

[JulienB]

Oh I think I just wrote the method wrong. If f'(x₀ ) = 0 and f''(x₀) < 0, then there is a maximum at x = x₀? That would make sense with the problem above.Julien.

 

[Ray Vickson]

Homework Statement

Hi everybody! It's kind of the first time that I try to solve this type of problem, so I'd like to see what you guys think about it and if I am heading in the right direction:

Examine if the set {(x² + x + 1)/(x² + 1) : x∈ℝ} has a maximum and/or a minimum, and calculate those extrema if so.

Homework Equations

Derivatives?

The Attempt at a Solution

Okay so I directly have a first question regarding the problem: can I treat the set as a function or is that nonsense? I think the derivatives of that expression should tell me where the slope of the function is zero and therefore potentially reaching an extremum (or not), right?

So here we go:

(d/dx)(x² + x + 1)/(x² + 1) = [(2x + 1)(x² + 1) - (x² + x + 1)(2x)]/(x² + 1)² = (-x² + 1)/(x² + 1)²

which is equal to 0 for x = {-1;1}. Now I take the second derivative of the set to check if it is also equal to 0 for x = {-1;1}, and if it is not then those values of x would mark two extrema.

(d/dx)(-x² + 1)/(x² + 1)² = [2x(x² - 3)]/(x² + 1)³

which is equal to 1/2 for x = -1 and -1/2 for x = 1, which would that the set has a minimum at x = 1 and a maximum at x = -1. But I know from the graph that that is wrong, although the values for x are right :/

Where is my mistake? Is that to begin with the right way to proceed? And if yes, are there also other methods? Do you have any remark about such problems, especially about the difference between extrema of sets and of functions?Thank you in advance for your answers.Julien.

Your conclusions are exactly backwards: $x = -1$ is a (global) minimum, while $x = 1$ is a global max.

 

[JulienB]

@Ray Vickson Yeah I realized that. Thank you for your answer.Julien.