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it a, b, c ER are such for all XFR, a t² + bx + (20₁ b². Yul Lo. Amit: Find the minimal value of the polynomial in t Show that that thin

it a, b, c ER are such for all XFR, a t² + bx + (20₁ b². Yul Lo. Amit: Find the minimal value of the polynomial in t Show that that thin

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Mathematics Exploration: Quadratic Inequality** *Problem Statement:* Show that if \( a, b, c \in \mathbb{R} \) are such that for all \( x \in \mathbb{R} \), \( ax^2 + bx + c \geq 0 \), then \( b^2 - 4ac \leq 0 \). *Hint:* Find the minimal value of the polynomial in \( x \). *Explanation:* This problem involves understanding the conditions under which a quadratic polynomial is always non-negative for all real values of \( x \). - **Concept**: A quadratic polynomial \( ax^2 + bx + c \), where \( a, b, c \) are real numbers, has a specific condition for having no real roots, which is when its discriminant \( b^2 - 4ac \leq 0 \). - **Objective**: Prove that if the given quadratic polynomial is non-negative for every real \( x \), then the above condition for the discriminant holds true. - **Approach**: 1. Conceptualize how the position of the vertex of the quadratic affects the entire graph. 2. Utilize the expression for the vertex to evaluate how it maintains non-negativity. 3. Apply the hint to find the minimal value using the vertex form or completing the square, linking to the condition \( b^2 - 4ac \leq 0 \). The underlying mathematics involves understanding the symmetry and properties of quadratic functions, ensuring the function doesn't dip below the x-axis, which is visually represented by this inequality condition.
Transcribed Image Text:**Mathematics Exploration: Quadratic Inequality** *Problem Statement:* Show that if \( a, b, c \in \mathbb{R} \) are such that for all \( x \in \mathbb{R} \), \( ax^2 + bx + c \geq 0 \), then \( b^2 - 4ac \leq 0 \). *Hint:* Find the minimal value of the polynomial in \( x \). *Explanation:* This problem involves understanding the conditions under which a quadratic polynomial is always non-negative for all real values of \( x \). - **Concept**: A quadratic polynomial \( ax^2 + bx + c \), where \( a, b, c \) are real numbers, has a specific condition for having no real roots, which is when its discriminant \( b^2 - 4ac \leq 0 \). - **Objective**: Prove that if the given quadratic polynomial is non-negative for every real \( x \), then the above condition for the discriminant holds true. - **Approach**: 1. Conceptualize how the position of the vertex of the quadratic affects the entire graph. 2. Utilize the expression for the vertex to evaluate how it maintains non-negativity. 3. Apply the hint to find the minimal value using the vertex form or completing the square, linking to the condition \( b^2 - 4ac \leq 0 \). The underlying mathematics involves understanding the symmetry and properties of quadratic functions, ensuring the function doesn't dip below the x-axis, which is visually represented by this inequality condition.
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