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Chapter 4: Problem 24

Show that the equation \(6 x^{5}+13 x+1=0\) has exactly one real root.

Short Answer

Expert verified
Given that \(f(x)\) is strictly increasing and changes sign from negative to positive, there is exactly one real root for the equation \(6 x^{5}+13 x+1=0\) as it crosses the x-axis exactly once.

Step by step solution

01

Writing the polynomial function

Write down the given function which is \(f(x) = 6x^5 + 13x + 1\).
02

Calculating the first derivative

The first derivative of the function can be calculated using basic differentiation rules. The derivative of \(x^n\) with respect to \(x\) is \(n*x^{(n-1)}\) and the derivative of a constant is zero. Thus the derivative of \(f(x)\) is \(f^{'}(x) = 5*6x^4 + 13 = 30x^4 + 13\).
03

Analyzing the first derivative

Observe that \(f^{'}(x)\) is always positive since \(30x^4 + 13 > 0\) for all \(x\), as the sum of a positive number and any real number squared is always positive and therefore the function is strictly increasing in all real line.
04

Analyzing the behavior of the function at infinity

Because the function is a polynomial of odd degree and the leading coefficient is positive, we know from calculus that \(f(x) \rightarrow -\infty\) as \(x \rightarrow -\infty\) and \(f(x) \rightarrow +\infty\) as \(x \rightarrow +\infty\).
05

Locating the root of equation

Because the function is strictly increasing and changes from negative to positive as \(x\) increases, it must cross the x-axis exactly once, at the root of the equation.

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Key Concepts

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

Strictly Increasing Functions
A function is described as strictly increasing when, for every pair of numbers, if one number is larger than the other, then the function's output for these numbers maintains this order鈥攊t always increases too.
Strictly increasing functions are vital because they ensure the uniqueness of solutions, especially in polynomial equations. If a polynomial function is strictly increasing, it cannot plateau or dip, and each point on the function corresponds to exactly one output.
  • For any values, say, \( a \) and \( b \), if \( a < b \), then \( f(a) < f(b) \).
  • This property implies that there are no stationary points where the derivative is zero or negative over the entire domain of \( x \).
  • Such functions guarantee that there is one and only one crossing with the x-axis, leading to exactly one real root for many polynomial equations.
First Derivative Test
The first derivative test is a technique used to determine the increasing or decreasing nature of a function by analyzing its first derivative.
The derivative, \( f'(x) \), of a function gives an instantaneous rate of change. When \( f'(x) > 0 \) across the entire domain, the function is strictly increasing.
  • Calculate the derivative of the function. For a polynomial function \( f(x) = 6x^5 + 13x + 1 \), the derivative is \( f'(x) = 30x^4 + 13 \).
  • Consistently positive values of \( f'(x) \) indicate the function increases without any decrease, plateau, or bending back over itself.
  • Verifying that \( 30x^4 + 13 > 0 \) for all real numbers shows that \( f(x) \) is strictly increasing.
This analysis confirms there is no real number for which \( f'(x) \leq 0\), thus maintaining strict monotonic behavior over the real line.
Polynomial Function Behavior at Infinity
Understanding how a polynomial behaves as you move towards infinity is crucial in determining the number of real roots.
The end behavior of a polynomial function is influenced by the degree and the leading coefficient of the polynomial.
  • The leading coefficient influences the vertical direction the function moves towards as \(x\) approaches infinity and negative infinity.
  • For odd-degree polynomials such as \( f(x) = 6x^5 + 13x + 1 \), the end behaviors will diverge: as \( x \rightarrow -\infty \), \( f(x) \rightarrow -\infty \); and as \( x \rightarrow +\infty \), \( f(x) \rightarrow +\infty \).
  • This divergent behavior ensures that the graph of the function crosses the x-axis exactly once, confirming a single real root.
Thus, understanding polynomial behavior at infinity, combined with the strictly increasing nature of the function, allows us to deduce the number of roots present.

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