Is a polynomial monotone when the first derivative has only imaginary roots?

Question

I have a polynomial over a specific the range. The first derivative has only two imaginary roots and no real roots. The first derivative is positive in the lower bound and upper bound. Does that mean that the function is monotone (increasing) over the range?

Answer 1

Not necessarily. The first derivative being positive at the lower and upper bounds of the range indicates that the function is increasing at those points. However, the function could still have other critical points within the range where it changes direction.

For instance, consider a polynomial with two imaginary roots within the specified range. While the first derivative is positive at the lower and upper bounds, there could be a local minimum or maximum within the range where the derivative is zero or undefined, causing the function to change direction.

To determine if the function is monotone over the entire range, you need to analyze the behavior of the first derivative and check for any additional critical points (where the derivative is zero or undefined) within the range. If there are no such critical points, then the function would indeed be monotone over that range.

Answer 2

Consider $f(x):=2x^3-9x^2+12x$. Then $f(x)$ has just one real zero, at $x=0$, the other two being complex. Also, its gradient $f'(x)=6(x^2-3x+2)=6(x-1)(x-2)$ is $12$ at $x=0$ and $x=3$, while it has a maximum at $x=1$ and a minimum at $x=2$. So it is not monotone in the range $0$ to $3$, while having a positive gradient at both ends of this range.

Answer to revised question: The derivative of a polynomial is also a polynomial. If this latter polynomial has only two imaginary roots and no real root, then it is either a quadratic or (if multiple roots are allowed) a power of a quadratic. In the real domain, a quadratic function that never takes the value zero is either always positive or always negative. So, in this case, the gradient is always positive. That is, the original polynomial is strictly increasing (and so monotone) everywhere.