Question

what is the maximum number of relative extrema contained in the graph of this function F(x)=5x^4-x^3+6x-2

Answer 1

Relative extrema occur where the derivative is zero (at least for your polynomial function).
So taking the derivative we get
\[20x^3-3x^2+6=0\]
This is a 3rd degree equation, now if we are working with complex numbers this equation is guaranteed to have 3 solutions by the fundamental theorem of algebra. But the number of real roots are 1 which can be found out by using Descartes' rule of signs ( https://en.wikipedia.org/wiki/Descartes'_rule_of_signs).
So the maximum number of relative extrema are 1.

Answer 2

Also that one must be a global minimum since the polynomial is of degree 4 and has a positive coefficient, so at large distances from the origin the function tends to infinity.