There can be small differences of terminology, but nothing as radical as what you are experiencing.
The function $f(x)$ has a relative maximum (also known as a local maximum) at $x=c$ if there is a positive $\epsilon$ such that $f(x)\le f(c)$ for all $x$ in the interval $(c-\epsilon, c +\epsilon)$.
Sometimes an exception is made when we are maximizing a function over a closed interval $[a,b]$. If there is an $\epsilon$ such that $f(x)\le f(a)$ for all $x$ in the interval $[a,a+\epsilon)$, then some people may say there is a relative maximum at $x=a$. I believe that most standard calculus books do not count endpoint maxima as relative maxima. Do check your text. Whatever it says is, for your purposes, the local dialect. That dialect may change a little when you get to university.
A relative maximum can be an absolute maximum. An absolute maximum can be a relative maximum. If we have an absolute maximum at $x=c$, and our function is defined in some interval $(c-\epsilon, c+\epsilon)$, then the absolute maximimum is automatically a relative maximum.
From your description, the exam is closest to using the words in the standard way. The text may also be substantially correct.
Remark: I am sorry that you are getting mixed messages. The online courses, at least in the Dogwood province, can be on the weak side.
Answer: The maximum number of strict local minima of a quartic polynomial is $N=5$.
A few examples of such polynomials are provided in the answers to this cross-posting: https://mathoverflow.net/questions/442736, and they are visualized at the end of this post. In what follows will be argued that $N$ can not be greater than 5.
The papers "Counting Critical Points of Real Polynomials in Two Variables" http://www.jstor.org/stable/2324459 and "The Index of grad f (x, y)", Alan H. Durfee, Topology Vol. 37, No. 6, pp. 1339Ð1361, 1998
shows that the number of critial points is bounded by the vector field index $i$ of the gradient vector field of the polynomial p(x,y).
Due to Bezout's theorem we know that we have at most 9 critical points (for an example, see the plot below).
The papers show that for non-degenerate critical points we have:
$$i=m+n-s$$ where $i$ is the index, $m$ is the number of local maxima, $n$ is the number of local minima, and $s$ is the number of saddle points.
The paper "The Index of grad f (x, y)" shows that: $$i \leq max(1,d−3),$$ where $d$ is the degree of the polynomial, in this case $d=4$ so that we get the bound $i \leq 1$.
Combining this with the index formula above we have as the most extreme possible case 5 minima, 0 maxima, and 4 saddle points, for a total of 9 critical points:
$$i=5+0-4=1$$
We cannot have more than 5 minima since then we don't get enough saddle points to get the index down to the bound 1. So, 5 is a definite upper bound. This is a polynomial with 4 local minima, 1 local maximum, and 4 saddle points for a total of 9 critical point and index 1:
$$p(x,y)=(8x^4-8x^2+1)+(16y^4-12y^2+1)$$
The OP gave a simpler polynomial on this form but I like this one since it is the sum of two Chebyshev polynomials.
Below is a plot of this polynomial where the red curves are the loci of the zeros of the gradient components. Where they intersect we have a critical point.
There are of course also degree 4 polynomials with 3 minima (and of course 2 and 1). Here is an example with 3 minima and 2 saddle points:
$$p(x,y)=(xy + y + x^2 -1)^2+(x^2 + 2xy + y^2 + x-y -1)^2$$
Finally, the cross-posting:
https://mathoverflow.net/questions/442736 features a few polynomials with 5 minima. Below is a visualization of one of the polynomials, discovered by Peter Mueller:
Note: the plot for (0,0) is rotated 180 degrees vs. the larger figure and some of the others are also rotated.
Here is a visualization of a simpler polynomial, found a bit later by Peter Mueller:
Note: The coordinate values are rounded and the orientation of each highlighted minimum has been changed for clarity. In this case the symmetry of the polynomial was used to show the nature of the minima (only 3 types). (The previous polynomial is also symmetric with 3 different types of minima but there all 5 are shown separately.)
Here is a zoomed-in view of the centrally located saddle point (located at the origin):
The visualization below shows yet another of Mueller's polynomials. This one with an interesting minimum at the origin, located close to a saddle point (see also https://mathoverflow.net/questions/442736):
The following visualization shows a polynomial written as a sum of squares with an interesting minimum at the origin (Mueller):
Note: This doesn't look like a minimum but it is! There are shallow saddle points on either side of the minimum.
Best Answer
B) is impossible as the shape of such graph does not matches with the behavior at $\pm$ infinities of a odd degree polynomial.
C) is possible as you can see here
Also this might be useful for you.