/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 55 Let \(f(x)=a x^{2}+b x+c,\) wher... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 55

Let \(f(x)=a x^{2}+b x+c,\) where \(a>0 .\) Prove that \(f(x) \geq 0\) for all \(x\) if and only if \(b^{2}-4 a c \leq 0 .\) [Hint: Find the minimum of \(f(x) .]\)

Short Answer

Expert verified
The function \( f(x) \geq 0 \) for all \( x \) if and only if \( b^2 - 4ac \leq 0 \).

Step by step solution

01

Understanding the Quadratic Function

A quadratic function of the form \( f(x) = ax^2 + bx + c \) is a parabola. Since \( a > 0 \), it opens upwards, which means it has a minimum point.
02

Find the Minimum of the Function

To find the minimum point of the quadratic function \( f(x) = ax^2 + bx + c \), we can use the vertex formula for the x-coordinate, \( x = -\frac{b}{2a} \).
03

Calculate the Minimum Value

Substitute \( x = -\frac{b}{2a} \) into \( f(x) \):\[f\left(-\frac{b}{2a}\right) = a\left(-\frac{b}{2a}\right)^2 + b\left(-\frac{b}{2a}\right) + c.\]Simplifying gives:\[f\left(-\frac{b}{2a}\right) = \frac{b^2}{4a} - \frac{b^2}{2a} + c = \frac{b^2}{4a} - \frac{2b^2}{4a} + c = \frac{b^2 - 4ac}{4a}.\]
04

Determine the Condition for Non-Negativity

For \( f(x) \geq 0 \) for all \( x \), the minimum value \( \frac{b^2 - 4ac}{4a} \) must be non-negative. Therefore, \( \frac{b^2 - 4ac}{4a} \geq 0 \) simplifies to \( b^2 - 4ac \leq 0 \).
05

Conclusion

Hence, the original statement is proven: The function \( f(x) = ax^2 + bx + c \) satisfies \( f(x) \geq 0 \) for all \( x \) if and only if \( b^2 - 4ac \leq 0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parabola
A quadratic function appears as a parabola on a graph. This form of equation, written as \( f(x) = ax^2 + bx + c \), has a distinct U-shape. When the coefficient \( a \) is greater than zero, the parabola opens upwards. This upward opening is crucial because it signifies that there is a minimum point somewhere along the curve. The lowest point of this upward-opening parabola is called the vertex. Understanding the nature of a parabola is key. Here are some important features to remember about parabolas:
  • If \( a > 0 \), the parabola opens upwards and the vertex represents a minimum point.
  • The axis of symmetry passes through the vertex and divides the parabola into two mirror images.
  • The shape and width of the parabola depend on the value of \( a \). A larger \( a \) makes the parabola narrower, while a smaller \( a \) makes it wider.
Knowing these properties will help you visualize and solve problems involving quadratic functions with confidence.
Vertex Formula
To pinpoint the minimum or maximum position of a parabola, you need to find its vertex. For a quadratic function \( f(x) = ax^2 + bx + c \), the vertex formula is essential. The x-coordinate of the vertex is calculated with the formula \( x = -\frac{b}{2a} \). Once this x-value is found, it can be substituted back into the original equation to find the corresponding y-coordinate, which gives the minimum or maximum value of the function. Here's a step-by-step:
  • Identify the coefficients \( a \), \( b \), and \( c \) in the quadratic expression.
  • Apply the vertex formula \( x = -\frac{b}{2a} \) to find the x-coordinate of the vertex.
  • Substitute \( x = -\frac{b}{2a} \) into the function \( f(x) \) to determine the minimum or maximum value (the y-coordinate).
    • This means finding \( f\left(-\frac{b}{2a}\right) \).
Understanding how to use the vertex formula is crucial to solving many problems involving quadratic equations. It provides a direct method to locate the peak or valley of the parabola.
Discriminant
The discriminant is an integral part of analyzing quadratic equations. It is calculated using the formula \( b^2 - 4ac \), found inside the quadratic formula \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \). The value of the discriminant tells us about the nature of the roots of a quadratic equation and helps in determining the characteristics of the parabola:
  • If the discriminant is positive \((b^2 - 4ac > 0)\), the quadratic equation has two distinct real roots and the parabola intersects the x-axis at two points.
  • If the discriminant is zero \((b^2 - 4ac = 0)\), the parabola touches the x-axis at one point. This means there is exactly one real root, and the vertex sits on the x-axis.
  • If the discriminant is negative \((b^2 - 4ac < 0)\), the equation has no real roots and the parabola does not intersect the x-axis. In such cases, the graph of the function stays above or below the x-axis, depending on the direction in which the parabola opens.
For the specific problem we're considering, the statement \( b^2 - 4ac \leq 0 \) indicates that the quadratic function \( f(x) = ax^2 + bx + c \) is always non-negative, or remains above the x-axis. This is essential for understanding which solutions pertain to continuous non-negativity.

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