inflection points of functions question If a function has no minima or maxima points, does it also mean in has no inflection points? since if no such points exist, it doesnt have to change concavity anywhere, so it sounds as a logical conclusion 
 A: In general you could have a smooth function that is strictly monotonic (either increasing or decreasing) that has infinitely many inflection points. Just consider drawing something like a smoothed out staircase plot where each horizontal step face becomes a graph that has an inflection point.
A: Firstly, I don't want to be a scrooge but it's better to use "local minima" and "local maxima"
Consider the function:
$$f(x) = x^3 + x$$
Clearly this function does not have a local minima or maxima (graph it if you don't believe me, but its sufficient to note that $x^3$ and $x$ are both monotonically increasing functions which are added constructively together), yet it has an inflection point! Note: 
$$f''(x)=6x$$
which has  zero at $x=0$. Now, mechanically, the criterion for an inflection point has been satisfied, but what does this all even mean.
Fundamentally a zero of the second derivative tells us that the first derivative has a critical point itself, be that a minima or a maxima. This implies that for the original function, there is a corresponding point where the slope starts changing the way it is changing! So if the slope it curving up, then it, after the inflection point, starts curving down. Lets say we have an x-y graph where x is for x-length and y is for y-length. We have a curve on this graph which refers to the motion of a point/"ball". Now just consider the curve a mapping o the point motion through this 2-dimensional space. So the ball's path begins with a u-shape (as shown below), very much like the round part a quadratic. Note: the ball must be beginning its motion with a downwards tilt, as it moves in that direction. The ball dips and at first, its trajectory is flattening out, then it temporarily moves at parallel to the x-axis and then it starting going upwards. However, the ball ball has been continually curving upwards, or view it this way, it has ceaselessly been turning to the "left"/upwards. But suddenly it starts curving down again, imitating the shape of a sine-curve. But of this to be possible, there must have been a point where it stopped turning upwards and started turning downwards! This point is called the inflection point! 
Here is a graph showing the above motion:

Now here is an even more bizarre movement graph. This time the ball, also begins with the u shaped motion, turning upwards, but before it becomes parallel with the x-axis, it starts turning downwards! 

What is important here? Well the fact that, you can have an inflection point before the slope actually ever crosses 0 / flatness. 
So forget the maxima/minima criterion, for you can have a function with an inflection point without any adjacent "flat" slopes!
The originally stated $x^3 + x$ has an inflection point at x=0, yet does not have a minima or maxima. In fact that's not it: it does not even have any points where the slope is 0/flat! 
So in general: 
A function can haven no minima or maxima and yet have an inflection point ($x^3$). 
But even the stronger criteria of not having a "flat" point is too weak. So in general: 
A function can have no "flat" bits and still have an inflection point ($x^3+x$)
The only sure way to check if a function has an inflection point is to check its second derivative.
I highly recommend you graph different functions and their corresponding derivatives in order to develop an intuition for what this all truly means.
Please ask my any questions you have!
