Question

Let a,b,c be non-zero real numbers such that b2<4ac, and f(x)=ax2+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 setthe 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.