# Discriminant for monotonic functions

NOTE: This is a part of a homework question and I do not intend to seek direct answers or don't wish to receive any. I have clearly mentioned what my confusion is in the last line(highlighted) of the description. That is all I need.

Given function $$f: \frac{x^3}{3} + \frac{a}{2}x^2 + bx + 10$$ where $$a$$ and $$b$$ is from the set $$A=\{1,2,3,4,5\}$$

I was asked to find the number of ordered pairs (a,b) from set A so that f is injective.

I did the following:

1. Found the 1st derivative and found the Discriminant of the quadratic( the derivative)

2. Now I took the Discriminant(say $$D$$) as $$D<0$$ as I thought for a monotonic function there wont be any sort of local maxima or minima in the graph which means, there is no point on the graph where the slope of the tangent is $$0$$.

3. So on evaluating $$b^2-4ac<0$$, I got the answer as $$13$$ possible ordered pairs by trial.

Now my homework answer key gives $$15$$ as the correct answer as they (the author of the book) took $$D\ge 0$$. Can someone please explain why the author took $$D\ge0$$. Just this part.

• Your choice of notation is a bit confusing to me, since I believe that the letters $a, b, c$ you are using in 3. are different from the ones you are using as coefficients for the polynomial. What you probably want to investigate is the term $a^2 - 4b$ with $a, b$ taken from the coefficient of the polynomial (there is no $c$ there, either). As for your question: I believe with the choice of notation I just made you want to have $a^2 - 4b \le0$. Note that $x\mapsto x^3$, as an example, is injective, even though it's derivative has a zero. This is true in general for third order polynomials. Feb 1, 2023 at 18:54