How to decide if a function is increasing (strictly) at point where derivative is zero It is easy enough to formally prove that $f(x) =x^3$ is increasing even when its derivative at $x=0$ is $0$ but what about some complex function,  say the polynomial $g(x)=
(3/2)x^4-3x^2+1$? Is it increasing,  decreasing or neither at $x=-1, 0,1$? (where its derivative is $0$). If it is increasing or decreasing then what is the meaning of zero rate of change of $y$ wrt $x$ in this case? 
Please answer a bit rigorously even though I am taking Calculus. 
 A: The notion "increasing" always refers to pairs of points: A function $f$ is increasing in the interval $J$ if for all $x<y$ in $J$ we have $f(x)\leq f(y)$, and is strictly increasing if for all $x<y$ in $J$ we have $f(x)< f(y)$. Therefore it makes no sense to ask whether a function "is increasing at $x=0$".
Now a familiar criterion for "increasing" is the sign of the derivative $f'$. One proves: If $f$ is differentiable in the interval $J$ then $f$ is increasing in $J$ iff $f(x)\geq0$ for all $x\in J$, and is strictly increasing iff for all $x\in J$ one has $f(x)\geq0$  but  in no subinterval of positive length $f(x)\equiv0$.
Isolated zeros of the derivative therefore do not destroy the monotonicity of $f$.
A: Keep taking derivatives until the first non-zero derivative at the point of interest. So for $x^3$:
$$f'(x)=3x^2$$
$$f''(x)=6x, f''(0) = 0$$
$x^3$ is peculiar, it has an inflection point here so take another derivative.
$$f'''(x) = 6, f'''(0) = 6$$
So, $x^3$ is an increasing function at $x=0$.
$g(x)$ next:
$$g'(x) = 6x^3 - 6x$$
As you already figured, at $-1$, $0$, and $1$, the first derivative is 0. Take another derivative:
$$g''(x) = 18x^2 - 6$$
$g''(0) = -6$ which means it curves down, it is a decreasing function at this point. $g''(-1) =g''(1) = 12$. It curves up at these points so it is increasing. If you plot $g(x)$ you can see this is true.
This has nothing to do with 'monotonically increasing' which is where it has to be increasing everywhere as in the $x^3$ function.
