Question 9

# Let a, b, c be non-zero real numbers such that $$b^2 < 4ac$$, and $$f(x) = ax^2 + bx + c$$. If the set S consists of all integers m such that f(m) < 0, then the set S must necessarily be

Solution

$$b^2 < 4ac$$ means that the discriminant is less than 0. Therefore, f(x)>0 for all x if the coefficient of $$x^2$$ is positive, and f(x)<0 for all x if the coefficient of $$x^2$$ is negative.

We are given that f(m)<0 and m is an integer.

So the set containing values of m will either be empty if the coefficient of $$x^2$$ is positive, or it will be a set of all integers if the coefficient of $$x^2$$ is negative.

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