For the following statement A implies B, write the statement NOT B implies NOT A.
Let $a$, $b$, and $c$ be real numbers. If $a > 0$, then there does not exist a real number $M$ such that, for every real number $x$, $ax^2 + bx + c \le M$.
If there exists a real number $M$ such that, for every real number $x$, $ax^2 + bx + c \le M$, then $a \le 0$.
If there exists real numbers $M$ and $x$ such that $ax^2 + bx + c > M$, then $a \le 0$.
As I understand it, NOT [for every] and NOT [$ax^2 + bx + c \le M$] should be equivalent to [there exists] and [$ax^2 + bc + c > M$]; indeed, this has been the case with all previous similar questions.
I'm wondering what I've misunderstood here, or is my solution correct?
I would greatly appreciate it if people could please take the time to clarify this.