Is y=x^3 Monotonic at x=0?

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In summary, the function y=x^3 is not strictly decreasing at x=0 because its derivative is 0 at that point. However, it is strictly decreasing at all other points. This is not a paradox, as the function is continuous and has only one point where the derivative is 0. The theorem stating that a strictly monotonic function has an inverse function with a derivative at every point is incorrect, as shown by a counter example.
  • #1
player1_1_1
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Homework Statement


[tex]y=x^3[/tex] is it monotonic in 0?

The Attempt at a Solution


if i try to solve [tex]y'>0[/tex] it will be 0 in point [tex]x=0[/tex] so that function is not strictly decreasing in 0 but in other way we have [tex]x_1<x_2\Rightarrow f(x_1)<f(x_2)[/tex] so it is scrictly decreasing, its paradox, why?
 
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  • #2
Why is y' = 0 not allowed for a stricly increasing function?
Isn't [tex]x_1 < x_2 \implies f(x_1) \le f(x_2)[/tex] the definition of monotonic (or strictly monotonic, if you replace the equality by a strict equality)?
 
  • #3
player1_1_1 said:

Homework Statement


[tex]y=x^3[/tex] is it monotonic in 0?

The Attempt at a Solution


if i try to solve [tex]y'>0[/tex] it will be 0 in point [tex]x=0[/tex] so that function is not strictly decreasing in 0 but in other way we have [tex]x_1<x_2\Rightarrow f(x_1)<f(x_2)[/tex] so it is scrictly decreasing, its paradox, why?

Plug in a value before 0 and after 0 in [tex]y'=3x^2[/tex]. You'll see that y' is strictly positive for all real values except 0, where the slope is zero. So it follows that this is neither a minimum nor maximum, and 0 is the only point that has slope zero. We also know that y is a continuous function, so y must be strictly increasing, i.e. monotone.
 
  • #4
so generally if [tex]x_0[/tex] is isolated point where derivative is 0 and its positive around this function can be strictly monotonic there yeah? and what about this theorem what i found? if function is scrictly monotonic in every point then function then inverse function has derivative in any point, but function [tex]y=\sqrt[3]{x}[/tex] doesn't have derivative in 0, why?
 
  • #5
Because the derivative of the function is not defined at x=0. The best you can hope for is a limit.
 
  • #6
player1_1_1 said:
and what about this theorem what i found? if function is scrictly monotonic in every point then function then inverse function has derivative in any point

Either you have misunderstood something about the statement of the theorem, or it is just wrong.

Counter example:
f(x) = -1 + x when x < 0
f(x) = 0 when x = 0
f(x) = 1 + x when x > 0
f(x) is strictly increasing, but you can't even define a limit of f^-1(0), let alone a derivative, because f^-1(x) is not defined when -1 < x < 0 and 0 < x < 1
 
  • #7
[itex]a^3- b^3= (a- b)(a^2+ ab+ b^2)= (a- b)((a^2+ ab+ b^2/4)+ (3/4)b^2)= (a-b)((a+b/2)^2+ 3/4 b^2)[/itex]

Since [itex]a^2+ ab+ b^2[/itex] is the sum of two squares it is never negative and [itex]a^3- b^3[/itex] is positive if and only if a- b is.
 

What does it mean for a function to be monotonic?

A monotonic function is one that either always increases or always decreases as its input variable increases. In other words, the function is either entirely "upward sloping" or "downward sloping".

How do I determine if a function is monotonic?

To determine if a function is monotonic, you can graph the function and see if it always increases or decreases. Alternatively, you can take the derivative of the function and see if it is always positive or always negative.

Can a function be both increasing and decreasing?

No, a function cannot be both increasing and decreasing. It must be one or the other to be considered monotonic.

Is a constant function considered monotonic?

No, a constant function, where the output is the same for all input values, is not considered monotonic as it does not have a consistent direction of change.

Why is it important to know if a function is monotonic?

Knowing if a function is monotonic can be useful in various applications, such as optimization problems, where we want to find the maximum or minimum value of a function. Monotonic functions also have nice properties that can make them easier to work with in mathematical calculations.

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