Final answer:
A polynomial of even order always has either a global maximum or a global minimum, but not both.
Step-by-step explanation:
A polynomial of even order always has either a global maximum or a global minimum, but not both.
This can be explained by the behavior of even functions. An even function is symmetric with respect to the y-axis, meaning that if you reflect one side of the graph over the y-axis, it will look exactly like the other side.
If the leading coefficient of the polynomial is positive, then the graph will open upwards and have a global minimum. If the leading coefficient is negative, then the graph will open downwards and have a global maximum. Since even functions are symmetric, they cannot have both a global maximum and a global minimum.