Problem 60
Question
Suppose that \(p(x)\) is a polynomial, and \(\left|p^{\prime \prime}(x)\right| \leq 1\) for all \(x\). Prove that \(p\) is of degree at most \(2,\) and the coefficient of its degree 2 term is less than or equal to \(1 / 2\) in absolute value.
Step-by-Step Solution
Verified Answer
The degree of \(p(x)\) is at most 2, and its quadratic coefficient satisfies \(|a| \leq 1/2\).
1Step 1: Understanding the Conditions
We begin by interpreting the given condition \(|p''(x)| \leq 1\) for all \(x\). This tells us that the second derivative of the polynomial is bounded by \(1\) everywhere, which limits the curvature or change in slope of the polynomial.
2Step 2: Analyzing the Degree of p(x)
Since \(p(x)\) is a polynomial and its second derivative \(p''(x)\) is bounded, there are constraints on the polynomial's degree. Realize that if \(p(x)\) had a degree higher than 2, then \(p''(x)\) would have terms involving \(x\), contradicting the bounded derivative condition \(|p''(x)| \leq 1\) for all \(x\). This implies that \(p''(x)\) must be constant, hence \(p(x)\) is at most quadratic (degree 2).
3Step 3: Determining the Maximum Coefficient of the Quadratic Term
Express \(p(x) = ax^2 + bx + c\), where \(a\) is the coefficient of the quadratic term. Compute \(p''(x) = 2a\). The condition \(|p''(x)| \leq 1\) implies \(-1 \leq 2a \leq 1\). Solving for \(a\), we get \(-\frac{1}{2} \leq a \leq \frac{1}{2}\). Thus, the coefficient \(a\) of the quadratic term is at most 1/2 in absolute value.
Key Concepts
Second DerivativeBounded CurvatureQuadratic PolynomialCoefficient Analysis
Second Derivative
The second derivative of a function, denoted as \(p''(x)\), gives us crucial insights about the curvature of the original function, \(p(x)\). Curvature relates to how sharply a curve bends.
A second derivative that is constant or bounded implies that the curve is relatively consistent in its bending motion. For polynomials, if the high-order terms are restricted or behave in a specific controlled manner, they determine the behavior of the second derivative.
In this exercise, knowing that \(|p''(x)| \leq 1\) for all \(x\) indicates a stringent bound on how much the polynomial \(p(x)\) can "bend" as \(x\) varies. This is key to proving the properties of the polynomial.
A second derivative that is constant or bounded implies that the curve is relatively consistent in its bending motion. For polynomials, if the high-order terms are restricted or behave in a specific controlled manner, they determine the behavior of the second derivative.
In this exercise, knowing that \(|p''(x)| \leq 1\) for all \(x\) indicates a stringent bound on how much the polynomial \(p(x)\) can "bend" as \(x\) varies. This is key to proving the properties of the polynomial.
Bounded Curvature
Bounded curvature refers to the constraint on how much the curve of the polynomial can change. Here, the curvature of \(p(x)\) is defined by its second derivative \(p''(x)\). A tightly bound second derivative like \(|p''(x)| \leq 1\) controls the maximum "bending" away from a straight line.
Such a bound suggests that sharp turns or large oscillations within the polynomial are prohibited. For a polynomial, this implies that high-degree behaviors, which normally exhibit significant curvature change, are restricted, reinforcing that the degree of \(p(x)\) is limited to at most two.
Such a bound suggests that sharp turns or large oscillations within the polynomial are prohibited. For a polynomial, this implies that high-degree behaviors, which normally exhibit significant curvature change, are restricted, reinforcing that the degree of \(p(x)\) is limited to at most two.
Quadratic Polynomial
A quadratic polynomial is a polynomial of degree two and takes the form \(p(x) = ax^2 + bx + c\). When dealing with the core concept of quadratic polynomials, it is essential to recognize that they represent parabolas, which may open upwards or downwards depending on the sign and magnitude of the coefficient \(a\).
In this exercise, the notion of the polynomial having at most degree two fits with the constraint \(|p''(x)| \leq 1\), implying that \(p''(x)\) must be a constant. Thus, quadratic polynomials are ideal candidates as they allow for limited curvature due to their inherent structure.
In this exercise, the notion of the polynomial having at most degree two fits with the constraint \(|p''(x)| \leq 1\), implying that \(p''(x)\) must be a constant. Thus, quadratic polynomials are ideal candidates as they allow for limited curvature due to their inherent structure.
Coefficient Analysis
Coefficient analysis involves evaluating the specific constants within a polynomial expressions. In our quadratic polynomial, \(p(x) = ax^2 + bx + c\), we are most interested in the coefficient \(a\) of the \(x^2\) term because it directly influences \(p''(x) = 2a\).
Given \(|p''(x)| \leq 1\), we deduce that \(-1 \leq 2a \leq 1\), simplifying to \(-\frac{1}{2} \leq a \leq \frac{1}{2}\). This exploration of \(a\) illustrates the power of coefficient analysis in understanding the effects on the curvature limits imposed on the polynomial and confirms that the quadratic coefficient remains moderate in magnitude.
Given \(|p''(x)| \leq 1\), we deduce that \(-1 \leq 2a \leq 1\), simplifying to \(-\frac{1}{2} \leq a \leq \frac{1}{2}\). This exploration of \(a\) illustrates the power of coefficient analysis in understanding the effects on the curvature limits imposed on the polynomial and confirms that the quadratic coefficient remains moderate in magnitude.
Other exercises in this chapter
Problem 60
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