Question

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

• A. the set of all integers
• B. either the empty set or the set of all integers
• C. the empty set the set of all positive integers
• D. the set of all positive integers

Answer 1

The Correct Option is B

Given the quadratic function $$f(x) = ax^2 + bx + c$$ where $$a, b,$$ and $$c$$ are non-zero real numbers such that $$b^2 < 4ac$$, we want to find the set $$S$$ of all integers $$m$$ such that $$f(m) < 0$$.
 The quadratic function $$f(x)$$ represents a parabola. The discriminant $$\Delta$$ of the quadratic equation 
$$ax^2 + bx + c = 0$$ is given by 
$$\Delta = b^2 - 4ac$$. 
Since $$b^2 < 4ac$$, the discriminant is negative ($$\Delta < 0$$), which means that the quadratic equation has two distinct complex roots. 
The vertex of the parabola is given by the point $$(h, k)$$ where $$h = -\frac{b}{2a}$$ and $$k = f(h)$$. 
In this case, since $$a$$ is non-zero, the parabola opens upwards if $$a > 0$$ and downwards if $$a < 0$$. 
Since the parabola opens upwards or downwards and the discriminant is negative, the parabola does not intersect the x-axis. This implies that the function $$f(x)$$ is either entirely above the x-axis (if $$a > 0$$) or entirely below the x-axis (if $$a < 0$$). 

Now, we want to find the set $$S$$ of all integers $$m$$ such that $$f(m) < 0$$. 
Depending on the sign of $$a$$, the parabola is either above or below the x-axis.
 If the parabola is above the x-axis, there will be no integer $$m$$ for which $$f(m) < 0\, (f(m)$$ will always be positive or zero). 
If the parabola is below the x-axis, then $$f(m) < 0$$ for all integers $$m$$. 

Therefore, the set $$S$$ must be either the empty set (if $$a > 0$$ and the parabola opens upwards) or the set of all integers (if $$a < 0$$ and the parabola opens downwards), depending on the sign of $$a$$. 
Hence, the correct answer is: either the empty set or the set of all integers.

Answer 2

The discriminant is smaller than zero if \(b ^2 <4ac\).
Consequently, in the event that the coefficient of x² is positive, \(f(x)>0\) for all x, and in the event that it is negative, \(f(x)<0\) for all x. 
It is given to us that m is an integer and that \(f(m)<0\). Therefore, if the coefficient of x² is positive, the set containing values of m will either be empty, or if the coefficient of x² is negative, it will consist of a set of all integers.