Can there be two relative minimums on the graph of a quartic function?

Question

I need to find the relative minimum and maximum of the given function:

$$x^4-4x^3-2x^2+5x+9$$

Graphing this, I get a relative maximum of $(0.538, 10.6)$ But, I seem to get two relative minimums, at $(-0.728,6.12)$ and $(3.19,-21.7)$

Which point is the relative minimum? Can there be two?

Answer 1

If by "relative minimum", you mean "local minimum", then yes, you can have two minimums, since the derivative of the quartic polynomial is of order three and can have three roots.

Your particular polynomial has two local minimums and one maximum, as seen on this graph

Answer 2

"Relative minimum" means the same thing as "local minimum." A function could have infinitely many local minima.

On the other hand, there can be at most one global minimum, though it could be attained at more than one point.