Problem 45 - Calculus Early Transcendentals: Pearson New International Edition

Question

Consider \(f(x)=A x^{2}+B x+C\) with \(A>0 .\) Show that \(f(x) \geq 0\) for all \(x\) if and only if \(B^{2}-4 A C \leq 0\).

Step-by-Step Solution

Answer

\( f(x) \geq 0 \) for all \( x \) if and only if \( B^2 - 4AC \leq 0 \).

Step 1: Identify the Function to Analyze

The given function is a quadratic function: \( f(x) = A x^2 + B x + C \), where \( A > 0 \). We need to establish conditions under which this function is always non-negative (\( f(x) \geq 0 \) for all \( x \)).

Step 2: Use the Discriminant

For a quadratic equation \( ax^2 + bx + c = 0 \), the discriminant \( \Delta = b^2 - 4ac \) is key in determining the nature of its roots. Specifically, if \( \Delta \leq 0 \), the equation has no real roots, or it has a repeated root. In this context, \( f(x) \geq 0 \) implies that the function does not intersect the x-axis more than once.

Step 3: Set \( f(x) \geq 0 \) Condition

For the parabola to be always non-negative, it must not cross the x-axis, which happens when the discriminant \( B^2 - 4AC \leq 0 \). This means either the quadratic has a double root, which implies a tangent at its vertex, or no real roots at all, hence never dipping below the x-axis.

Step 4: Establish Final Conclusion

We conclude that for \( f(x) \geq 0 \) for all \( x \), it is necessary and sufficient that \( B^2 - 4AC \leq 0 \). This condition ensures the parabola opens upwards (since \( A > 0 \)) and does not intersect the x-axis at separate points.

Key Concepts

Discriminant

The discriminant is a special expression used with quadratic equations to reveal important information about their roots. For a general quadratic equation, written as \( ax^2 + bx + c = 0 \), the discriminant is given by the formula \( \Delta = b^2 - 4ac \). This value is crucial because it tells us whether the equation has real roots:

• If \( \Delta > 0 \), the quadratic equation has two distinct real roots.
• If \( \Delta = 0 \), the equation has exactly one real root, sometimes called a repeated or double root.
• If \( \Delta < 0 \), there are no real roots, meaning the roots are complex or imaginary.

In the context of the function \( f(x) = A x^2 + B x + C \), with \( A > 0 \), if \( B^2 - 4AC \leq 0 \), the equation has at most one real root. This results in the quadratic function not intersecting or merely touching the x-axis. It supports the condition that the graph of the function remains above or on the x-axis, fulfilling the non-negative function requirement.

Non-negative Function

A non-negative function is one that does not take on any negative values for its entire domain. For the quadratic function \( f(x) = A x^2 + B x + C \), where \( A > 0 \), proving that it is non-negative for all \( x \) involves checking its relationship with the x-axis. The function is non-negative if its graph never dips below the x-axis.

For a parabola to be non-negative, it implies:

• The vertex of the parabola lies at or above the x-axis.
• The parabola opens upwards, which is already given because \( A > 0 \).
• The discriminant \( B^2 - 4AC \) must be less than or equal to zero, indicating at most one contact with the x-axis, either as a tangent or not at all.

By ensuring these conditions, the function \( f(x) \geq 0 \) holds true for all \( x \), meaning it is always non-negative.

Roots of Quadratic Equation

The roots of a quadratic equation represent the values of \( x \) where the function equals zero, i.e., where \( ax^2 + bx + c = 0 \). For our specific quadratic function \( f(x) = A x^2 + B x + C \) with \( A > 0 \), determining the roots gives insight into where the function might cross the x-axis.

The roots are determined by the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]The solutions derived from this formula depend heavily on the discriminant \( b^2 - 4ac \):

• If the discriminant is positive, there are two distinct roots, and the graph of the quadratic crosses the x-axis at two points.
• If the discriminant is zero, there's exactly one root, where the graph touches the x-axis at the vertex.
• If the discriminant is negative, there are no real roots, and the graph does not touch or cross the x-axis at all.

When \( B^2 - 4AC \leq 0 \) for our quadratic function, it confirms that the graph does not dip below the x-axis, satisfying the condition where the function remains non-negative.