Question of calculus concepts

Question

Today I learned some concepts of increasing and decreasing functions, but some is so vague, so I have many questions. Here I can have a list of questions:

1. Functions f(x) is decreasing on an interval (a,b) then f'(x)<0 on (a,b) yes? But what if f'(x)<0. Is a function f(x)decreasing if $f'(x)<0$ on an interval (a,b)
2. Functions f(x)>f(b) in the interval (a,b). so is f decreasing on (a,b).
3. Monotonic functions are flat. If it is a random function f(x), if f(a)=f(b), then is f(x) monotonic on the interval (a,b)?

Thanks

Answer 1

1. If a function is decreasing it means its slope is negative, i.e., $f^\prime (x) < 0$.

2. No. $f(x) > f(b)$ does not mean that $f(x)$ is decreasing. It could go up and down (so long as it is always greater than $f(b)$.

3. No. Monotonic means that the function is either always increasing or always decreasing.

Answer 2

1. Decreasing does not imply $f’(x)<0$. For example, consider the function $f(x)=-x^3$ at the point $(0,0)$. $f’(0) = 0, $ however the function is strictly decreasing, even at $x=0$. Also constant functions like $f(x)= 4$ are decreasing...However the converse is true: $f’(x) < 0 \implies$ strictly decreasing.
2. No. Consider $\sin x$ on $[\frac\pi6,\pi]$
3. Monotonic functions are not necessarily constant functions. But if $f(a)=f(b)$ and $f$ is monotonic on $[a,b]$ then this forces $f$ to be constant. However, I think you could do with clarification. See: https://math.stackexchange.com/a/3907156/29156