Determine if a function is strictly increasing/decreasing

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In summary: However, if X has at least two points, then these statements are false because there will be a choice of x and y for which the implication reduces to "true implies false", which is a false statement.In summary, the conversation discusses a function and its derivative, as well as its domain and extreme points. It also raises a question about whether the function can be strictly increasing and strictly decreasing at the same point. The answer is that it depends on the definition of "strictly increasing at x".
  • #1
tompenny
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Homework Statement
Determine where a function is strictly increasing/decreasing
Relevant Equations
$$f(x)=x+\frac{1}{(x+1)}$$
Hi there.

I have the following function:

$$f(x)=x+\frac{1}{(x+1)}$$

I've caculated the derivative to:

$$f'(x)=1-\frac{1}{(1+x)^2}$$

And the domain to: $$(-\infty, -1)\cup(-1, \infty)$$

I've also found two extreme point: $$x=0, x=-2$$

I know that a function is strictly increasing if:
$$f'(x)> 0$$
and strictly decreasing if:
$$f'(x)< 0$$

I've calculated the intervalls where the funtcion is strictly increasing to:
$$(-\infty, -2]\cup[0, \infty)$$
and strictly decreasing to:
$$[-2, -1)\cup(-1, 0]$$

My question is if this is correct or if the intervalls should be:
$$(-\infty, -2)\cup(0, \infty)$$ and $$(-2, -1)\cup(-1, 0)$$ instead?
As you can notice I'm very unsecure whether I should use ( or [ at the extremum points?

Any help would be greatly appreciated.
Thank you:)

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  • #2
Can a function be strictly increasing and strictly decreasing at the same point?
 
  • #3
I really don't know.. are you reffering to the extremum point at x=-2?
 
  • #4
tompenny said:
I really don't know.. are you reffering to the extremum point at x=-2?
Yes. In your first solution you have the two points where ##f'(x) = 0## (i.e. ##x = 0## and ## x = -2##) as both strictly incraesing and strictly decreasing. That makes no sense.

Your second solution has got to be right.
 
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  • #5
PeroK said:
Can a function be strictly increasing and strictly decreasing at the same point?

It depends on what you mean by "strictly increasing at [itex]x[/itex]".

A function is strictly increasing or decreasing on some subset of its domain. It can't be strictly increasing or decreasing on a subset consisting of a single point, because the definitions don't really make sense in that case.[1]

Here we have:

It is true that [itex]f[/itex] is strictly increasing on [itex][0,\infty)[/itex].
It is true that [itex]f[/itex] is strictly decreasing on [itex](-1,0][/itex].

On the other hand, if by "[itex]f[/itex] is strictly increasing at [itex]x[/itex]" you mean "there is an open neighbourhood of [itex]x[/itex] on which [itex]f[/itex] is strictly increasing" then [itex]f[/itex] is not strictly increasing or decreasing at 0.

[1]: If [itex]X[/itex] consists of a single point then [tex](\forall x \in X)(\forall y \in X) ((x < y) \implies (f(x) < f(y)))[/tex] and [tex](\forall x \in X)(\forall y \in X) ((x < y) \implies (f(x) > f(y)))[/tex] are true, because whatever the choice of [itex]x[/itex] and [itex]y[/itex] the implications reduce to "false implies false", which is a true statement.
 
Last edited:

1. What is a strictly increasing/decreasing function?

A strictly increasing function is a mathematical function where the output (y-value) increases as the input (x-value) increases. In other words, as the input increases, the output also increases. A strictly decreasing function is the opposite, where the output decreases as the input increases.

2. How do you determine if a function is strictly increasing/decreasing?

To determine if a function is strictly increasing/decreasing, you can graph the function and see if it has a positive (increasing) or negative (decreasing) slope. Alternatively, you can take the derivative of the function and see if it is always positive (increasing) or always negative (decreasing).

3. Can a function be both strictly increasing and strictly decreasing?

No, a function cannot be both strictly increasing and strictly decreasing. This is because a strictly increasing function has a positive slope, while a strictly decreasing function has a negative slope. Therefore, the slope cannot be both positive and negative at the same time.

4. What is the difference between strictly increasing/decreasing and non-decreasing/non-increasing functions?

A strictly increasing/decreasing function has a positive/negative slope, meaning that the output always increases/decreases as the input increases. On the other hand, a non-decreasing/non-increasing function can have a zero slope, meaning that the output can remain constant even as the input increases.

5. How can knowing if a function is strictly increasing/decreasing be useful?

Knowing if a function is strictly increasing/decreasing can be useful in many real-world applications, such as analyzing trends in data or predicting future outcomes. It can also help in solving optimization problems, where the goal is to find the maximum or minimum value of a function.

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