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

The condition \( B^2 - 4AC \leq 0 \) ensures that \( f(x) \geq 0 \) for all \( x \).

Step 1: Understanding the Function

We have a quadratic function \( f(x) = Ax^2 + Bx + C \). The coefficient \( A \) is positive (\( A > 0 \)), meaning the parabola opens upwards. To determine when \( f(x) \) is non-negative for all \( x \), we need to analyze its roots and vertex.

Step 2: Condition for Non-negative Function

For \( f(x) \geq 0 \) for all \( x \), the quadratic must have a minimum point at or above the x-axis. This occurs when the discriminant \( B^2 - 4AC \) is non-positive, meaning the quadratic has either no real roots or a double root.

Step 3: Deriving the Discriminant Condition

The discriminant \( \Delta = B^2 - 4AC \) indicates the nature of the roots. When \( \Delta \leq 0 \), the parabola does not intersect the x-axis, or it is tangent, indicating \( f(x) \) does not change sign and remains non-negative.

Step 4: Conclusion on the Given Condition

We verify that the condition \( B^2 - 4AC \leq 0 \) ensures that the minimum value of the parabola is 0 or positive, confirming \( f(x) \geq 0 \) for all \( x \). Thus, the statement \( f(x) \geq 0 \) if and only if \( B^2 - 4AC \leq 0 \) holds true.

Key Concepts

Discriminant

In the world of quadratic equations, the discriminant plays a crucial role in determining the nature of the roots. It is the part of the quadratic formula under the square root: \( \Delta = B^2 - 4AC \). The value of the discriminant gives insight into the type of solutions the quadratic equation will have.
- **Positive Discriminant (\( \Delta > 0 \))**: The quadratic equation has two distinct real roots. The parabola intersects the x-axis at two points.
- **Zero Discriminant (\( \Delta = 0 \))**: The quadratic equation has exactly one real root, meaning the parabola is tangent to the x-axis. This is equivalent to having a double root.
- **Negative Discriminant (\( \Delta < 0 \))**: The quadratic equation has no real roots. Instead, the solutions are complex conjugate pairs. The parabola does not intersect the x-axis at all.
Understanding the discriminant helps us predict the quadratic's behavior, such as whether it stays non-negative across all values of \( x \).

Quadratic Function

A quadratic function is usually written in the form \( f(x) = Ax^2 + Bx + C \), where \( A \), \( B \), and \( C \) are constants, and \( A eq 0 \). This function represents a parabola in a cartesian coordinate system.
When \( A > 0 \), the parabola opens upwards, creating a "U" shape. This orientation is crucial when determining whether the function is non-negative.
The axis of symmetry can be found using \( x = -\frac{B}{2A} \), and the vertex, which is the highest or lowest point depending on the direction of the parabola, provides insight into its minimum or maximum value.
Key Features of Quadratic Functions:

• The vertex can either be the minimum or maximum of the function depending on the sign of \( A \).
• The parabola will have a minimum point when \( A > 0 \), and we need to check if this point is on or above the x-axis for non-negative conditions.
• The discriminant determines the nature of roots, influencing whether the parabola crosses the x-axis.

Quadratic functions model a wide range of real-life scenarios, making understanding their properties essential.

Non-negative Function

A function is considered non-negative if it has no negative values for all inputs. For the quadratic function \( f(x) = Ax^2 + Bx + C \), ensuring \( f(x) \geq 0 \) for all \( x \) implies that it never dips below the x-axis. When the quadratic function's parabola opens upwards (as \( A > 0 \)), the whole graph must stay on or above the x-axis.
To guarantee this:

• The discriminant \( \Delta = B^2 - 4AC \) must be less than or equal to zero. This ensures the function does not have two distinct real roots where it might go negative.
• When \( \Delta = 0 \), the function has a minimum value exactly at the x-axis, touching it but not crossing below.
• If \( \Delta < 0 \), the function has no real roots and remains entirely above the x-axis.

Therefore, the condition \( B^2 - 4AC \leq 0 \) ensures that the quadratic function is non-negative across its domain, fulfilling the requirement that \( f(x) \geq 0 \) for all \( x \).