Question

function f is defined for \[x \neq 0\]
\[f(x)= \frac{ 2x ^{2} }{ x-1 }\]

determines the asymptotes of the graph of f
determined intervals and growth and degrowth relative extrema.

Answer 1

If you have a function defined on:
\[\forall t \notin \left\{ t_0 \right\}\]
Then you have:
\[\frac{df}{dx} > 0 \]
means INCREASING (as the slope, "rate of change", is positive) and if it is less than zero it is DECREASING. If it is ZERO you have a horizontal tangent line (which normally indicates a max or min but doesn't have to)

For:
\[\frac{d^2 f}{dx^2}>0\]
you have the function is CONCAVE UP and if it is less than zero you have CONCAVE DOWN. If it is zero then you have an "inflection point" or a point where the concavity changes. For example if we have:
\[f(x)=x^2 \rightarrow f'(x)=2x \rightarrow f''(x)=2\]
We see for x LESS than zero the function is DECREASING (f'(x<0)=-2|x|<0) and for x GREATER than zero the function is INCREASING (f'(x>0)=2x>0). For the second derivative we see it is constant and POSITIVE which means the function is concave up for all x. To test our intuition lets graph this function:

Answer 2

All of this of course holds for:
\[t \ne t_0\]