Why does this function have a saddle point ? $f(x)=3x^4+4x^3$ I have this function
$$f(x)=3x^4+4x^3$$
I have to find the extreme points. I did but just to be sure I put the problem in an online calculator and it said saddle point (0; 0). After some research I noticed that saddle points are common for two variables functions.. but I don't have a two variables function so why is there a saddle point ? 
Wolfram says it's an inflection point but Symbolab says it's a saddle point. Can someone enlighten me please ? 
Thanks in advance.
 A: That's nonsense. Saddle points are named as such because of their shape as part of a two-dimensional surface, and they have no lower-dimensional analogue. Wolfram is correct in calling it an inflection point.
After seeing this, I wouldn't trust Symbolab at all!
A: As already pointed out, a function of a single variable does not have a saddle point; rather, perhaps, you are alluding to a point of inflection---a point where the graph of the function changes concavity.
Candidates for such points are points for which $f''(x) = 0$.
Here, $f''(x) = 0$ occurs when $x = -1$ or $x = 0$.
To check if any of these yields a point of inflection, we check the concavity of the function over the intervals $(-\infty, -1)$, $(-1, 0)$ and $(0, \infty)$ by taking suitable test points a determining the sign of $f''(x)$.
Let's check, say, $x=-2$, $x = -1/2$ and $x=1$.
Observe, since $f''(-2)<0$, $f''(-1/2)>0$, and $f''(1)>0$, the graph of $f(x)$ changes concavity only through the candidate $x = -1$.
Hence, $(-1, -1)$ is the only point of inflection.  
