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Chapter 4: Problem 19

Show that the cubic \(2 x^{3}+3 x^{2}+6 x+10\) has exactly one real root.

Short Answer

Expert verified
To show that the given cubic function \(2x^3 + 3x^2 + 6x + 10\) has exactly one real root, we first examine the coefficients and find no sign changes, implying no positive real roots. We then check for negative real roots by replacing \(x\) with \((-x)\) and find one sign change, indicating one negative real root. Analyzing the monotonicity of the function with its derivative \(f'(x) = 6x^2 + 6x + 6\), which is always positive, we conclude the function is strictly increasing. As the function takes negative values for small negative \(x\) and positive values for large positive \(x\), it must cross the x-axis exactly once, indicating the existence and uniqueness of the real root.

Step by step solution

01

Examine sign changes in coefficients

We are given the cubic function: \(2x^3 + 3x^2 + 6x + 10\). Examine the coefficients: \(2, 3, 6, 10\) which are all positive. According to Descartes' Rule of Signs, there are no sign changes indicating there are no positive real roots for the given cubic function.
02

Investigate negative real roots

We know the function has no positive real roots, so we will now investigate about the negative real roots. To check for the negative roots, we substitute \(x\) with \((-x)\) in the given cubic function: \[2(-x)^3 + 3(-x)^2 + 6(-x) + 10\] Simplifying the expression, we get: \[-2x^3 + 3x^2 - 6x + 10\] Examine the coefficients of this modified cubic function: \(-2, 3,-6, 10\). We see that there are three sign changes, indicating that there is exactly one negative real root for the given cubic function.
03

Analyze the monotonicity of the function

To analyze the monotonicity, we need to find the derivative of the given cubic function. The derivative helps us understand the behavior of the function and its likely turning points. The derivative of the given cubic function, \(2x^3 + 3x^2 + 6x + 10\), with respect to \(x\) is: \[f'(x) = 6x^2 + 6x + 6\] Observe that the coefficients of the derivative are all positive. Hence, the derivative is always positive for real values of \(x\). This implies that the function is strictly increasing throughout its domain.
04

Existence and uniqueness of the real root

Since the function is increasing throughout, it must cross the x-axis at most once. To prove the existence, we need to show that the function takes both positive and negative values, ensuring that it does intersect the x-axis. For a small enough negative value, \(x < 0\), we observe that the function becomes: \(2x^3 + 3x^2 + 6x + 10 \approx 2(\text{-abs}(x))^3 < 0\) The function takes negative values. For large enough positive values (e.g., \(x = 1\)), we observe that the function becomes: \(2x^3 + 3x^2 + 6x + 10 \approx 2(1)^3 + 3(1)^2 + 6(1) + 10 > 0\) The function takes positive values as well. Since the function is strictly increasing and takes both positive and negative values, it must have a unique real root. In conclusion, through the inspection of coefficients' sign changes, analysis of the monotonicity, and demonstration of the existence and uniqueness of the real root, we have shown that the given cubic function \(2x^3 + 3x^2 + 6x + 10\) has exactly one real root.

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Key Concepts

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

Cubic Functions
Cubic functions are polynomial functions of degree three, generally represented as \(ax^3 + bx^2 + cx + d\). In these expressions, \(a\), \(b\), \(c\), and \(d\) are constants, with \(a eq 0\). The shape of the graph of a cubic function is diverse and can include:
  • Two turning points
  • One inflection point
  • A possible local maximum and minimum
Cubic functions may not always intersect the x-axis, but they can have up to three real roots. Since cubic functions are continuous and have no gaps, their overall behavior over the domain depends mostly on the sign and size of the coefficients. Understanding these basic properties helps to determine the number of real roots and their nature.
Descartes' Rule of Signs
Descartes' Rule of Signs provides a method to estimate the number of positive and negative real roots of a polynomial.
For positive real roots, you count how many times the sign of the coefficients changes as you read from left to right. Similarly, for negative real roots, you substitute each occurrence of \(x\) with \(-x\) and then count the sign changes.
In the exercise, the polynomial \(2x^3 + 3x^2 + 6x + 10\) has all positive coefficients. No sign changes means no positive real roots are possible. When substituting \(-x\), the modified polynomial \(-2x^3 + 3x^2 - 6x + 10\) shows three sign changes, which suggests there could be up to three negative real roots. However, not all of them must be real due to intermediate cases of complex roots emerging in conjugate pairs.
Monotonic Functions
A function that is either entirely non-increasing or non-decreasing throughout an interval is known as a monotonic function. For identifying if a function is monotonic, derivatives come in handy.
In particular, the derivative of a function helps determine its increasing or decreasing nature. When the derivative is positive across the domain, the function is strictly increasing.
For \(f(x) = 2x^3 + 3x^2 + 6x + 10\), its derivative is \(f'(x) = 6x^2 + 6x + 6\). Since all terms in the derivative are positive for any real value \(x\), the function is strictly increasing over its entire domain. This means there is only one continuous way the function can intersect with the x-axis, again pointing to the existence of a single real root.
Uniqueness of Real Roots
Understanding the uniqueness of real roots involves showing that only one point exists where a strictly increasing or decreasing function can cross the x-axis. This is especially true if the function takes on both negative and positive values as you move from negative to positive x-values.
For the cubic examined, \(2x^3 + 3x^2 + 6x + 10\), values of the function change from negative for \(x<0\) to positive for \(x>0\). Due to its increasing nature, it can only cross the x-axis once, confirming the presence of a singular real root.
This uniqueness condition stems from the continuous nature of polynomial functions and their strict increasing or decreasing behavior derived from their first derivatives. Proof of real root existence, alongside monotonicity, solidifies the understanding of a unique crossing with the x-axis.

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Most popular questions from this chapter

Find values of the constants \(a, b\), and \(c\) for which the graphs of the two functions \(f(x)=x^{2}+a x+b\) and \(g(x)=x^{3}-c, x \in \mathbb{R}\), intersect at the point \((1,2)\) and have the same tangent there.

Consider the following application of L'H么pital's Rule: $$ \lim _{x \rightarrow 1} \frac{3 x^{2}-2 x-1}{x^{2}-x}=\lim _{x \rightarrow 1} \frac{6 x-2}{2 x-1}=\lim _{x \rightarrow 1} \frac{6}{2}=3 $$ Is it correct? Justify.

Define \(f: \mathbb{R} \backslash\\{1\\} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\) by \(f(x):=1 /(x-1)\) for \(x \in \mathbb{R} \backslash\\{1\\}\) and \(g(x):=x\) for \(x \in \mathbb{R}\). Show that \(\frac{f^{\prime}(x)}{g^{\prime}(x)} \rightarrow-\infty\) as \(x \rightarrow 1^{+}, \quad\) but \(\quad \frac{f(x)}{g(x)} \rightarrow \infty\) as \(x \rightarrow 1^{+}\) Does this contradict L'H么pital's Rule? Justify.

Let \(f:[a, b] \rightarrow \mathbb{R}\) be such that \(f^{\prime}\) is continuous on \([a, b]\) and \(f^{\prime \prime}\) exists on \((a, b)\). Show that \(f^{\prime \prime}(c)(f(b)-f(a))=f^{\prime}(c)\left(f^{\prime}(b)-f^{\prime}(a)\right)\) for some \(c \in(a, b) .\)

Use the definition of a derivative to find \(f^{\prime}(x)\) if (i) \(f(x)=x^{2}, x \in \mathbb{R}\) (ii) \(f(x)=1 / x, 0 \neq x \in \mathbb{R}\) (iii) \(f(x)=\sqrt{x^{2}+1}, x \in \mathbb{R}\) (iv) \(f(x)=1 / \sqrt{2 x+3}, x \in(-3 / 2, \infty)\)
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