In this section we want to explore how the leading term effects global extrema. To this end, let’s initially consider two monomials; $p(x) = x^3$ and $q(x) = x^4$. If we plug in some positive numbers, they seem similar (although $q(x)$ does increase much faster), but the negative values are where the more noteworthy differences lay. Consider the following table of values;

Notice that $p(x)$ has both positive and negative values, whereas $q(x)$ has only positive values. After some consideration one can probably see that it’s because the even power of $x$ in $q(x)$ is eliminating the negative sign for any negative imput, whereas the odd power of $p(x)$ does not. Specifically, odd powers preserve the negatives, whereas even powers annihilate them.

So for $p(x) = x^3$, plugging in a large positive $x$ value yields a large (and still positive) output. On the other hand, if we were to use a large negative $x$ value, we would get a large (and negative) $y$ value as the output. But this means that, no matter what value we think of, a big enough positive or negative input will yield a more positive or more negative output. In other words, $p(x)$ won’t have a global maximum or minimum, because we can always just take a larger positive number to overcome any proposed maximum number, or larger negative numbers to overcome any proposed minimum number.

However, for $q(x) = x^4$ we can see that both large positive and large negative numbers $x$ values will yield large positive $y$ value outputs. This means that, on the one hand there is no maximum value, but on the other hand this also means there must be a minimum somewhere, because the $y$ value output will never get large and negative.

We should also recall that, if we use a negative coefficient, it flips the overall function over the $x$ axis; so maximums become minimums and minimums become maximums. Thus $p(x) = -x^3$ still doesn’t have a max or min, but $q(x) = -x^4$ would have a maximum somewhere.

The general result is that a polynomial (with a domain of all real numbers) whose leading term has an odd power can’t have any global max or min, but if the leading term has an even power, then it has a global minimum if the leading coefficient is positive, and a global maximum if the leading coefficient is negative.