Can an inflection exist if there's no max/min? Very quick question: if a function doesn't have a maximum nor minimum, can it still have a point of inflection? 
I believe that these two go hand in hand and without one you can't have the other but I just want to verify.
 A: A point of inflection is where concavity changes. The function $x^3$ has an inflection point, and no absolute or relative maxima or minima.
For an example where furthermore the derivative is nowhere $0$, we can use $x+x^3$.
A: consider a saddle point (the green $\color{green}{dot}$) for functions of 2 variables.

Suppose we took a knife and cut the graph down vertically at the green point. This would generate a curve. The derivative of the curve at the green point (no matter how we cut the graph) would be zero, but the second derivative would be zero, negative, or positive (depending upon how the cut was made).
This is a case where some slices may have a local extrema, but the entire function is unbounded with no local extrema.
Consider the Saddle Dome in Calgary (its a misnomer because its actually an hyperbolic paraboloid)

the red curve is concave upwards; the blue is concave downwards, and the green and orange lines are actually straight lines.
For the single variable case, see André Nicolas's answer and comments; he did a great job of explaining.
A: Possible cases may happen when we have a continuous function $f(x)$ and$$f'(x_0)=0,f''(x_0)=0$$ wherein $x_0$ be an inflection point of the function.
