91Ó°ÊÓ

(a) Suppose that the polynomial function \(f(x)=x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}\) has exactly \(k\) critical points and \(f^{\prime \prime}(x) \neq 0\) for all critical points \(x\). Show that \(n-k\) is odd. (b) For each \(n,\) show that if \(n-k\) is odd, then there is a polynomial function \(f\) of degree \(n\) with \(k\) critical points, at each of which \(f^{\prime \prime}\) is non-zero. (c) Suppose that the polynomial function \(f(x)=x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}\) has \(k_{1}\) local maximum points and \(k_{2}\) local minimum points. Show that \(k_{2}=k_{1}+1\) if \(n\) is even, and \(k_{2}=k_{1}\) if \(n\) is odd. (d) Let \(n, k_{1}, k_{2}\) be three integers with \(k_{2}=k_{1}+1\) if \(n\) is even, and \(k_{2}=k_{1}\) if \(n\) is odd, and \(k_{1}+k_{2} < n .\) Show that there is a polynomial function \(f\) of degree \(n,\) with \(k_{1}\) local maximum points and \(k_{2}\) local minimum points. Hint: Pick \(a_{1} < a_{2} < \cdots< a_{k_{1}+k_{2}}\) and try \(f^{\prime}(x)=\prod_{i=1}^{k_{1}+k_{2}}\left(x-a_{i}\right) \cdot\left(1+x^{2}\right)^{l}\) for an appropriate number \(l.\)

Step by step solution

01

Understand the First Derivative

Given the polynomial function \[f(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_0\] the first derivative is \[f'(x) = n x^{n-1} + (n-1) a_{n-1} x^{n-2} + \cdots + a_1.\] Critical points occur where \(f'(x) = 0\).

02

Analyze the Sign of the Second Derivative

The second derivative, which is \[f''(x) = n (n-1) x^{n-2} + (n-1)(n-2) a_{n-1} x^{n-3} + \cdots + 2 a_2,\] is non-zero at all critical points, indicating that these points are either local maxima or minima.

03

Apply Rolle’s Theorem

By Rolle's Theorem, every interval between critical points must contain at least one root of \(f'(x)\), and between each pair of roots of \(f'(x)\), there is one change in the sign of \(f''(x)\). Thus, for \(k\) critical points, the polynomial degree \(n\) provides \(n-k\) intervals, making \(n-k\) necessarily an odd number.

04

Construct a Polynomial for Part (b)

To show that there exists a polynomial with given \(n\) and \(k\), consider \[f'(x) = \prod_{i=1}^k (x - a_i) \cdot (1 + x^2)^l,\] where \(a_1 < a_2 < \cdots < a_k\) and \(l\) is chosen such that the total degree is \(n-1\). Integrating \(f'(x)\) gives a polynomial \(f(x)\) of degree \(n\) with \(k\) critical points.

05

Analyze Local Maxima and Minima for Part (c)

For a polynomial of degree \(n\), if \(f\) has \(k_1\) local maxima and \(k_2\) local minima, then if \(n\) is even, \(f\) alternates between peaks and valleys, so there must be one more minimum than maximum, \(k_2 = k_1 + 1\). If \(n\) is odd, the number of maxima and minima are balanced, \(k_2 = k_1\).

06

Construct a Polynomial for Part (d)

Given \(n\), \(k_1\), and \(k_2\) with \(k_1 + k_2 < n\), consider \[f'(x) = \prod_{i=1}^{k_1+k_2} (x - a_i) \cdot (1 + x^2)^l\] with \(a_1 < a_2 < \cdots < a_{k_1+k_2}\). By choosing \(l\) such that the total degree is \(n-1\), this generates a polynomial \(f(x)\) of degree \(n\) exhibiting \(k_1\) local maxima and \(k_2\) local minima.

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Key Concepts

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

polynomial function

A polynomial function is an expression made up of variables and coefficients, structured in the form of sums of terms. Each term consists of a variable raised to a non-negative integer power and multiplied by a coefficient. For example, a polynomial of degree n can be written as:
\[f(x) = x^n + a_{n-1} x^{n-1} + \text{...} + a_0 \].
Here, each power of x is associated with a coefficient (e.g., \(a_{n-1}\), \(a_0\)). The degree of the polynomial (n) is determined by the highest exponent on the variable x.
In these polynomials, we’re often interested in the points where the slope of the function, or its rate of change, is zero. This is where the first derivative comes into play.

first derivative

The first derivative of a polynomial function represents the slope or the rate at which the function's value changes. Mathematically, if \( f(x) \) is a polynomial, its first derivative \( f'(x) \) is found using differentiation rules for each term:
\[ f'(x) = n x^{n-1} + (n-1) a_{n-1} x^{n-2} + \text{...} + a_1 \].

• The first derivative is used to identify critical points, where the derivative equals zero \(f'(x) = 0\).
• These points are significant because they represent possible local maxima, local minima, or points of inflection.

By finding the x-values where \( f'(x) = 0 \), we get the critical points of the polynomial function.

second derivative

The second derivative of a polynomial function provides information about the concavity of the function. It is found by differentiating the first derivative:
\[ f''(x) = n(n-1) x^{n-2} + (n-1)(n-2) a_{n-1} x^{n-3} + \text{...} + 2 a_2 \].

• If the second derivative \( f''(x) \) is positive at a critical point, the function is concave up, indicating a local minimum.
• If \( f''(x) \) is negative, the function is concave down, indicating a local maximum.
• If \( f''(x) = 0 \), further analysis is needed to determine the nature of the concavity.

Therefore, the second derivative helps us confirm whether a critical point is a local maximum or a local minimum.

Rolle's Theorem

Rolle’s Theorem is a fundamental result in calculus that states:
If a function \( f \) is continuous on the closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and \( f(a) = f(b) \), then there exists at least one point \( c \) in \((a, b)\) such that \( f'(c) = 0 \).

This theorem is particularly useful when analyzing polynomial functions because it guarantees the existence of a root for the first derivative between any two critical points. This suggests the polynomial's behavior and provides constraints that help us understand the relationship between the number of critical points and the polynomial's degree.

local maxima

A local maximum of a polynomial function is a point \( x \) where the function reaches a peak within a small neighborhood. Mathematically, \( f \) has a local maximum at \( x = c \) if:

• \( f'(c) = 0 \) (the slope is zero, indicating a critical point)
• \( f''(c) < 0 \) (the second derivative is negative, confirming concavity down)

Local maxima are important as they represent the highest value the function attains within a specific interval. They allow us to understand the function's behavior in real-world scenarios, such as optimizing a quantity or analyzing natural phenomena.

local minima

A local minimum of a polynomial function is a point \( x \) where the function reaches the lowest value within a small neighborhood. Mathematically, \( f \) has a local minimum at \( x = c \) if:

• \( f'(c) = 0 \)
• \( f''(c) > 0 \) (indicating concavity up)

Local minima are just as crucial as local maxima for understanding the function's behavior. They help in minimizing costs, reducing risks, or understanding valleys in natural terrains. Tracking local minima and maxima provides a full picture of the polynomial's critical behavior over its domain.