Problem 11 - Thomas Calculus

Question

Explain why the following four statements ask for the same information: i) Find the roots of \(f(x)=x^{3}-3 x-1\) ii) Find the \(x\) -coordinates of the intersections of the curve \(y=x^{3}\) with the line \(y=3 x+1\) iii) Find the \(x\) -coordinates of the points where the curve \(y=x^{3}-3 x\) crosses the horizontal line \(y=1\) iv) Find the values of \(x\) where the derivative of \(g(x)=\) \((1 / 4) x^{4}-(3 / 2) x^{2}-x+5\) equals zero.

Step-by-Step Solution

Answer

All statements require solving \( x^3 - 3x - 1 = 0 \).

Step 1: Analyze Statement (i)

The statement involves finding the roots of the polynomial function \( f(x) = x^3 - 3x - 1 \). The roots are the values of \( x \) for which \( f(x) = 0 \), meaning the values of \( x \) that satisfy the equation \( x^3 - 3x - 1 = 0 \).

Step 2: Analyze Statement (ii)

This statement requires us to find the \( x \)-coordinates where the curve \( y = x^3 \) and the line \( y = 3x + 1 \) intersect. This means setting the expressions for \( y \) equal to each other: \( x^3 = 3x + 1 \). Rearranging gives \( x^3 - 3x - 1 = 0 \), which is the same equation as in statement (i).

Step 3: Analyze Statement (iii)

The statement involves finding \( x \)-coordinates where the curve \( y = x^3 - 3x \) crosses the horizontal line \( y = 1 \). We set the expression for the curve equal to 1: \( x^3 - 3x = 1 \). Rearranging gives \( x^3 - 3x - 1 = 0 \), which is identical to the equation in statement (i).

Step 4: Analyze Statement (iv)

This statement asks us to find the values of \( x \) where the derivative of the function \( g(x) = \frac{1}{4}x^4 - \frac{3}{2}x^2 - x + 5 \) equals zero. Taking the derivative, we get \( g'(x) = x^3 - 3x - 1 \), and setting this equal to zero presents the equation \( x^3 - 3x - 1 = 0 \), matching the requirement in statement (i).

Step 5: Conclusion

Each statement ultimately requires solving the equation \( x^3 - 3x - 1 = 0 \). Thus, they all ask for the values of \( x \) that are roots of this equation.

Key Concepts

Roots of Polynomial Functions

Understanding the roots of polynomial functions is crucial when solving equations. A polynomial is an expression made with variables, exponents, and coefficients. For example, in our exercise, the polynomial \( f(x) = x^3 - 3x - 1 \) is what we are working with. The roots of this function are the values of \( x \) for which the polynomial equals zero, i.e., \( f(x) = 0 \).
The roots tell us the points where the graph of the polynomial will intersect the x-axis on a coordinate plane. This is because at these points, the y-value is zero. To find these roots, one might factor the polynomial or use methods such as the quadratic formula, synthetic division, or numerical approaches if the factors are not easily apparent. In our example, finding the roots means solving the equation \( x^3 - 3x - 1 = 0 \).

The concept of roots is foundational as it applies to many mathematical procedures, like intersecting graphs or solving dynamic models.

Intersection of Curves

The intersection of curves occurs when two functions meet at the same points in the coordinate system. In our scenario, consider two functions: a curve, \( y = x^3 \), and a line, \( y = 3x + 1 \).
To find the intersection points, we set the two expressions equal to each other: \( x^3 = 3x + 1 \). By rearranging the equation to \( x^3 - 3x - 1 = 0 \), it's clear that we are once again solving for the roots of the polynomial function from earlier, or where \( f(x) = 0 \).

Finding intersections of curves is vital in many fields such as physics, engineering, and economics because it can represent equilibrium points or solutions where two conditions are simultaneously satisfied. These can represent solutions to systems of equations or balance points in optimization problems.

Derivative Setting to Zero

In calculus, setting a derivative to zero is a method to find the stationary points of a function, which could be local minima, maxima, or points of inflection. In this exercise, we have the function \( g(x) = \frac{1}{4}x^4 - \frac{3}{2}x^2 - x + 5 \).
To find where the slope of the function is zero (flat), we compute its derivative: \( g'(x) = x^3 - 3x - 1 \). By setting \( g'(x) = 0 \), we identify points where the slope is zero, indicating potential maxima, minima, or saddle points.

This process links to our initial polynomial equation \( x^3 - 3x - 1 = 0 \). Setting the derivative to zero is an essential skill for sketching curves and solving optimization problems in calculus because it helps to pinpoint critical points on the function's graph.

Polynomial Root Analysis

Analyzing the roots of a polynomial involves more than just finding solutions to \( f(x) = 0 \). It also includes understanding the behavior of the polynomial around its roots. For \( x^3 - 3x - 1 \), the roots may be deduced using algebraic manipulation or numerical methods.
Employing tools like the Rational Root Theorem or graphing methods can help predict and verify roots visually. For complex polynomials, numerical techniques such as the Newton-Raphson method may be employed to estimate roots to a desired degree of accuracy.

Analyzing polynomial roots not only aids in solving equations but also helps in comprehending the nature and shape of polynomial graphs, particularly how they transition across the x-axis. Knowing how to determine and analyze roots is invaluable in diverse applications ranging from signal processing to models in economics.