Problem 52 - Calculus A New Horizon

Question

(a) Use a CAS to graph the function $$f(x)=\frac{x^{4}+1}{x^{2}+1} $$ and use the graph to estimate the \(x\) -coordinates of the relative extrema. (b) Find the exact \(x\) -coordinates by using the CAS to solve the equation \(f^{\prime}(x)=0\).

Step-by-Step Solution

Answer

Graph the function to estimate extrema; then find exact extrema by solving \(f'(x)=0\) with a CAS.

Step 1: Understanding the Function

We are given the function \(f(x) = \frac{x^{4} + 1}{x^{2} + 1}\). This function is a rational function where the numerator is a fourth-degree polynomial and the denominator is a second-degree polynomial.

Step 2: Graph the Function

Using a Computer Algebra System (CAS), plot the function \(f(x) = \frac{x^{4} + 1}{x^{2} + 1}\). Analyze the graph to visually estimate where the relative maxima and minima occur by looking for peaks and troughs.

Step 3: Estimate Relative Extrema from Graph

From the graph, we observe potential relative extrema. These are likely at the peaks and troughs seen in the plot. Record these \(x\)-coordinates for further verification.

Step 4: Differentiate the Function

To find the exact \(x\)-coordinates, differentiate the function using CAS. The derivative \(f'(x)\) helps identify critical points. Use the quotient rule: \( f'(x) = \frac{(x^2 + 1)(4x^3) - (x^4 + 1)(2x)}{(x^2 + 1)^2} \). Simplify this expression to obtain \(f'(x)\).

Step 5: Solve f'(x) = 0

Using the CAS, solve the equation \(f'(x) = 0\) to find the \(x\)-coordinates where the derivative is zero, which correspond to the critical points of the function. These are the points where relative extrema can occur.

Step 6: Analyze the Solutions

The solutions from the derivative tell us the exact \(x\)-coordinates of the relative maxima and minima. Verify these against the graph, ensuring consistency between estimated and calculated points.

Key Concepts

Rational Functions

Rational functions are expressions that involve the division of two polynomial functions. They are structured as fractions where both the numerator and the denominator are polynomials. In the case of the function \( f(x) = \frac{x^{4} + 1}{x^{2} + 1} \), you observe a fourth-degree polynomial in the numerator and a second-degree polynomial in the denominator.

Rational functions can display a wide range of behaviors on a graph. These behaviors include asymptotes, intercepts, and potential breaks or holes in the graph depending on the polynomials' roots. In this problem, we need to explore how the graph behaves by first plotting it, and then analyzing where significant changes occur, such as peaks or valleys, known as relative extrema.

Relative Extrema

Relative extrema refer to the peaks (maxima) and valleys (minima) within a graph of a function. These points are where the function achieves a local maximum or minimum value, compared to its nearby values.

To estimate the relative extrema of the function \( f(x) \), we can initially look at the graph generated by a Computer Algebra System (CAS). By visually inspecting the graph, we identify where the curve reaches its highest and lowest points relative to its surrounding regions. However, visual estimation only gives us an approximation.

• Relative maxima are the tops or peaks of a curve.
• Relative minima are the bottoms or troughs.

For precise identification, further mathematical steps involving derivatives must be taken as outlined in the problem.

Graphical Analysis

Graphical analysis involves examining a function's plot to identify key characteristics like relative extrema, symmetry, intercepts, and even general trends. When you graph rational functions using tools like a CAS, you can see the overall shape of the curve and any peculiar behaviors.

For the function \( f(x) = \frac{x^{4} + 1}{x^{2} + 1} \), graphical analysis allows estimation of key points that may not be immediately evident from the equation alone. By examining the graph:

• You can propose where relative maximum or minimum points might exist.
• You can detect intervals where the function is increasing or decreasing.

This step sets the stage for more detailed calculus-based calculations to ensure these visual observations hold mathematically.

Critical Points

Critical points on a graph of a function are where the function’s derivative is zero or undefined. They are potential locations for relative extrema (maxima or minima) or points of inflection.

In calculus, finding critical points involves differentiating the function and solving \( f'(x) = 0 \). This indicates places where the slope of the tangent to the graph is zero, implying a potential peak or trough. For the given function:

• Differentiating gives us \( f'(x) \), computed using the quotient rule due to the function's rational form.
• Solving \( f'(x) = 0 \) with the help of a CAS provides the exact \( x \)-coordinates of critical points.

These coordinates help confirm the locations of relative extrema initially estimated from graphical analysis.

Derivative

The derivative of a function reflects the rate of change of the function concerning one of its variables, typically \( x \). It is a foundational tool in calculus used to find slopes of tangent lines, solve optimization problems, and identify critical points.

For rational functions like \( f(x) = \frac{x^{4} + 1}{x^{2} + 1} \), computing the derivative involves the quotient rule, due to the division of two polynomials. The derivative helps gauge how the function behaves at each point, indicating increases, decreases, or leveling off points.

• The derivative \( f'(x) \) is computed as \( \frac{(x^2 + 1)(4x^3) - (x^4 + 1)(2x)}{(x^2 + 1)^2} \).
• Solving \( f'(x) = 0 \) pinpoints critical points where relative extrema may happen.

The derivative not only helps in finding relative extrema but also plays a crucial role in understanding the overarching dynamics of the function.