Answer by Alon Amit:

if a function — any function — has roots (equals 0 at some point), then it cannot be always-positive or always-negative, because 0 is neither positive nor negative.

In the other direction, if a continuous function isn't always-positive or always-negative, then it must have a root. That's because the function is positive somewhere, and negative elsewhere, and being continuous means that it must cross 0 somewhere in between.

Therefore, your quadratic polynomial will be always-positive or always-negative precisely when it has no (real) roots, which is precisely when its discriminant [math]\Delta=b^2-4ac[/math] is negative.

So:

- Negative discriminant: no roots, must be either positive everywhere or negative everywhere.
- Zero or positive discriminant: has roots, not positive everywhere nor negative everywhere.

In the first case, you also want to determine which of the positive/negative cases you're in. Easiest way is to look at the leading coefficient [math]a[/math]: if it's positive, your equation shoots off to positive infinity for both large and small values of [math]x[/math], so it's always positive. Vice versa for negative [math]a[/math].

Questions of positivity and negativity don't have a direct relationship with the derivative. Two quadratics can have the exact same derivatives everywhere, while one is positive everywhere and the other is not. Compare, for example, [math]x^2+1[/math] and [math]x^2-1[/math].

When is a function like ax^2+bx+c always positive or negative? And what is its relationship with derivative and discriminant?

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