Statement of the definition of decreasing/increasing function

Question

I recently encounter a problem considering the definition for decreasing and increasing function. Here is what it looks like:

Given a function $y =f(x)$ for all $x \in \mathbb{R}$, and a interval $(a,b)$, which is the right statement out of all below:

A. $y=f(x)$ is decreasing on $(a,b)$ if and only if $\forall x \in (a,b), f'(x) \leq 0$.

B. If $\forall x \in (a,b),f'(x) \leq 0$ then $y = f(x)$ is decreasing on $(a,b)$.

C. $y=f(x)$ is decreasing on $(a,b)$ if and only if $\forall x \in (a,b). f'(x) < 0$.

D. If $\forall x \in (a,b), f'(x) < 0 $ then $y = f(x)$ is decreasing on $(a,b)$.

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The problem with this question, as far as I am concerned of my own, is that I do not quite understand how those statement are formed and relate to each other. It is even more complicated for me, since given the material definition:

Suppose $f$ has derivative on $(a,b)$, then if $f(x)$ is decreasing on $(a,b) \Rightarrow f'(x) \leq 0, \forall x \in (a,b)$. Similarly, If $f'(x) < 0, \forall x \in (a,b) \Rightarrow$ $f(x)$ is decreasing on $(a,b)$.

It seems to me that I can not understand the logical meaning of those statements, thus I can't found out the difference between $f'(x) > 0$ and $f'(x) \geq 0$ in this case. Also considering that I have $f(x)$, what if the function look like this?

$\hspace{3cm}$

It is quite confusing for me to read the two definitions in a same line, and the way it is said also bugged me. Additionally, I kind of feel like I am misunderstanding at some point of the difference between necessity statement and sufficient statement, since the way the logical relation is connected in the definition is exactly $P \rightarrow Q$ as far as I concerned. But still to me it is quite difficult to wrap my head around it. What I have to do in this situation?

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Hint:

The solution to the problem is answer D, though the text provided no detailed solution or explanation of this matter.

Answer 1

A is not correct because some decreasing functions may not be differentiable. For example sequence $f(n) = -n, n \in \mathbb{N}$.

B is not correct because $f'(x) \leq 0$ means either $f'(x) < 0$ or $f'(x) = 0$ is true. Let $f(x) = 0$ be a constant function, then it definitely satisfies $f'(x) = 0$, so it also satisfies $f'(x) \leq 0$, but $f$ is not decreasing.

C is not correct and the reason is the same as that of A.

Thus D is correct.

Answer 2

The distinction between the pairs A/C and B/D is just a matter of definition; when the relations involve strict inequalities, then we talk about strictly increasing/decreasing functions, while increasing/decreasing functions usually allow equality.

As for the simple or double implication, the double implication (i.e. the "if and only if" proposition) is not true, because a decreasing function, even when assumed continuous, is not necessarily differentiable. A simple counterexample would be given by $f(x) = -\sqrt[3]{x}$, which is strictly decreasing on all $\mathbb{R}$, but $f'(x) = -\frac{1}{3x^{2/3}}$ is not defined for $x \le 0 $.

In conclusion, as is, if I were you, I would choose the option B.