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Chapter 4: Problem 95

Can you have a finite absolute maximum for \(y=a x^{3}+b x^{2}+c x+d\) over \((-\infty, \infty)\) assuming \(a\) is non-zero? Explain why or why not using graphical arguments.

Short Answer

Expert verified
No, cubic functions diverge to infinity, so they cannot have a finite absolute maximum over \((- fty, + fty)\).

Step by step solution

01

Identify the Type of Function

The given function \(y = ax^3 + bx^2 + cx + d\) is a cubic polynomial, which is a degree 3 polynomial function. We are tasked with evaluating whether it can have a finite absolute maximum over the interval \((-fty, +fty)\).
02

Analyze the Behavior of Cubic Functions

Cubic functions have the general form \(ax^3 + bx^2 + cx + d\), where \(a eq 0\). Such functions extend infinitely in both directions on the y-axis if \(a\) is not zero. If \(a > 0\), the tails of the graph go down on the left and up on the right. Conversely, if \(a < 0\), the tails go up on the left and down on the right.
03

Determine the Limits of the Function

As \(x\) approaches \(-\infty\) or \(+\infty\), the dominant term in the function \(ax^3\) will dominate the behavior of the polynomial. Thus, if \(a > 0\), the function tends towards \(+\infty\) and \(-\infty\) on either end; if \(a < 0\), it approaches \(-\infty\) and \(+\infty\) respectively.
04

Conclusion Based on Graphical Behavior

Given that the cubic polynomial diverges to infinity in either direction depending on the sign of \(a\), there cannot be a finite absolute maximum on \((-fty, +fty)\). The function does not have a value where it achieves a maximum as it increases or decreases infinitely.

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Key Concepts

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

Polynomial Behavior
Cubic functions like the given function \( y=ax^3+bx^2+cx+d \) have interesting behaviors due to their polynomial nature. In polynomial behavior, it's important to understand how the function behaves as certain variables grow large or small.

A degree 3 polynomial, or cubic polynomial, is defined by its highest term, which involves the cube of the variable \( x \). This term \( ax^3 \) is critical because it ultimately determines how the polynomial behaves as \( x \) gets very small or very large.
  • If \( a > 0 \), the function will have different signs at \(-\infty\) and \(\infty\).
  • If \( a < 0 \), the signs will be reversed.
This information is crucial for understanding how cubic functions do not simply plateau or stay within any finite range.
Infinite Limits
Infinite limits describe what happens to a function as its input grows extremely large or small - essentially what happens at the extremes. For a cubic function where \( a eq 0 \), the dominant term \( ax^3 \) largely determines these limits.

As \( x \) approaches \( -\infty \) or \( +\infty \), this term dictates the rise or fall of the function as:
  • If \( a > 0 \), the function rises toward \( \infty \) as \( x \) goes to \( +\infty \) and falls toward \( -\infty \) as \( x \) goes to \( -\infty \).
  • If \( a < 0 \), the opposite is true: the function falls toward \( -\infty \) as \( x \) goes to \( +\infty \) and rises toward \( \infty \) as \( x \) goes to \( -\infty \).
Given this behavior due to infinite limits, a degree 3 polynomial cannot have a finite absolute maximum, as its values will continuously ascend or descend without bound as \( x \) advances in either direction.
Degree 3 Polynomial
The degree of a polynomial directly influences its behavior and graph. A degree 3 polynomial, whose highest degree term is \( x^3 \), can be thought of as the primary driver of the function's behavior.

This gives such polynomials a characteristic S-shape when graphed. Consider:
  • The influence of the cubic term \( ax^3 \), which dictates the overall direction the graph tails lean as \( x \) approaches extreme values.
  • Lower-degree terms, like quadratic or linear terms, affect the shape but not the infinite behavior of the tails.
The dominance of the cubic term is why, despite any local maxima or minima, the function cannot have an absolute maximum or minimum over \((-\infty, \infty)\) other than infinity.
Graphical Analysis of Functions
Graphical analysis is a fundamental method for interpreting functions visually. By graphing a cubic polynomial \( y=ax^3+bx^2+cx+d \), we gain insights into its behavior across all \( x \) values.

When performing a graphical analysis of cubic functions:
  • Observe the end behavior: this tells us the function's tendency at extreme \( x \)-values due to the cubic term \( ax^3 \).
  • Note any turning points: these are local maximas or minimas determined by derivatives but do not affect the end-behavior.
These graphical patterns confirm that while local extrema may exist, they do not equate to a finite absolute extremum over an infinite domain, reinforcing the limitless nature of cubic functions.

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Most popular questions from this chapter

Find the critical points and the local and absolute extrema of the following functions on the given interval. $$f(x)=x+\sin ^{2}(x) \text { over } [0, \pi]$$

For the following exercises, use both Newton鈥檚 method and the secant method to calculate a root for the following equations. Use a calculator or computer to calculate how many iterations of each are needed to reach within three decimal places of the exact answer. For the secant method, use the first guess from Newton鈥檚 method. $$f(x)=\sin x, x_{0}=1$$

Use the Mean Value Theorem and find all points \(0 < c < 2\) such that \(f(2)-f(0)=f^{\prime}(c)(2-0)\) $$ f(x)=\cos (2 \pi x) $$

Use Newton's method to find the first two iterations, given the starting point. $$y=x^{3}+1, x_{0}=0.5$$

Find the critical points and the local and absolute extrema of the following functions on the given interval. $$f(x)=3 x^{4}-4 x^{3}-12 x^{2}+6 \text { over }[-3,3]$$
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