Always increasing condition of a function In a classroom I was told today that the function is always increasing if $f'(X)\ge0$.
Interestingly my teacher took a function as $f(x)=x^3+3x^2+3x+5$ which is always increasing although at $x=-1$, derivative will be zero.
Now this equal to zero disturbed me a lot. Suppose there is a function which is increasing and then for some continuous interval it gets constant and after that interval it again starts increasing.
Now $f'(x)\ge0$ will be satisfied in such a case, but is such a function always increasing, obviously its no.
I tried to google it but couldn't find such a case.
What am I missing here?
 A: I think there's some confusion between a function $f$ being considered to just be "increasing", which means $f(x) \ge f(y)$ if $x \gt y$, and it being "strictly increasing", which means $f(x) \gt f(y)$ if $x \gt y$.
In your case, you're correct the function $f$ is not necessarily "strictly increasing" (e.g., if $f(x) = c$, with $c$ being a constant, then $f'(x) \ge 0$ but $f$ is not strictly increasing). Note that $f'(x) \ge 0$ only guarantees the function is "increasing". I believe this is what your teacher meant (but I'm not quite sure what was meant by "always"), even though their example of $f(x) = x^3 + 3x^2 + 3x + 5$ is actually strictly increasing. It's also possible they meant strictly increasing and made a mistake by not stating $f'(x) \gt 0$ instead. To be completely certain, I suggest you ask your teacher to clarify.
A: In a comment you wrote: "can there be multiple points in a function where its derivative is zero and still it is strictly increasing."
For the function $f(x) = x + \sin x,$ we have $f'(x) = 1 + \cos x,$ and this is $0$ at every odd multiple of $\pi,$ and yet the function is strictly increasing.
If $f'=0$ at an isolated point and $f'>0$ at every point in some open interval containing that point, then $f$ is strictly increasing on that interval. You can see that by drawing a picture.
Remeber that $\text{“}$strictly increasing$\text{''}$ is equivalent to saying the slope of every secant line is positive.
