Problem 73 - Calculus Early Transcendentals

Question

Use a CAS to graph \(f^{\prime}\) and \(f^{\prime \prime},\) and then use those graphs to estimate the \(x\) -coordinates of the relative extrema of \(f\). Check that your estimates are consistent with the graph of \(f .\) $$ f(x)=\frac{10 x^{3}-3}{3 x^{2}-5 x+8} $$

Step-by-Step Solution

Answer

Use CAS to graph \(f'(x)\) and \(f''(x)\), estimate extremas from zero crossings and sign changes, and verify with \(f(x)\)'s graph.

Step 1: Graph the First Derivative

To start, we will graph the first derivative of the function, denoted as \(f'(x)\). Use a computer algebra system (CAS) to compute \(f'(x)\) from the given function \(f(x)=\frac{10x^3-3}{3x^2-5x+8}\). Once the derivative is computed, use the CAS to plot this function. Look for places where the derivative is zero or undefined, as these are potential relative extrema.

Step 2: Graph the Second Derivative

Next, graph the second derivative \(f''(x)\). Use the CAS to find \(f''(x)\) from \(f'(x)\). Plot \(f''(x)\) on the CAS. Look for changes in sign in this graph, as this can indicate points of inflection where relative extrema may exist.

Step 3: Identify Potential Extrema from Graphs

Examine the graphs of \(f'(x)\) and \(f''(x)\) together. An \( x \)-coordinate of a relative maximum is where \( f'(x) \) changes from positive to negative (it crosses the x-axis from above), and \( f''(x) < 0 \) at that point. Similarly, an \( x \)-coordinate of a relative minimum is where \( f'(x) \) changes from negative to positive, and \( f''(x) > 0 \).

Step 4: Verify with the Original Function

Now, plot the original function \(f(x)\) using the CAS. Check these estimated \(x\)-coordinates against the points on the graph of \(f(x)\) to confirm they align with relative maxima or minima on the graph of \(f(x)\). Ensure the slopes around these points match the observations from the first and second derivative graphs.

Key Concepts

Graphing Derivatives

Graphing derivatives helps us understand how the original function behaves in terms of its increasing or decreasing nature. When you graph the first derivative, denoted as \(f'(x)\), you are essentially seeing where the slope of the tangent line to the curve goes to zero (a potential indication of a local maximum or minimum) or where the slope changes direction.

• If \(f'(x) > 0\), the function \(f(x)\) is increasing.
• If \(f'(x) < 0\), the function is decreasing.
• Where \(f'(x) = 0\), the function may have a relative extremum.

By using a Computer Algebra System (CAS), it's possible to quickly visualize these aspects through digital graphing, giving an immediate visual indication of where critical points for extrema might be located.

Relative Extrema

Relative extrema refer to points on a graph where a function reaches a local high or low in its immediate vicinity, meaning these are the "peaks" (maximums) or "valleys" (minimums) on the graph. These can be identified by observing where the derivative of the function changes sign.

• A relative maximum occurs where \(f'(x)\) changes from positive to negative.
• A relative minimum is found where \(f'(x)\) changes from negative to positive.

Relative extrema are crucial in understanding the topology of the graph as they highlight significant changes in the function's increase or decrease.

Computer Algebra System (CAS)

A Computer Algebra System, or CAS, is a powerful tool for performing symbolic mathematics. It can handle complex algebraic calculations, derivatives, integrals, and even graphical representations of functions and their derivatives.
With CAS, instead of manually computing derivatives, you can quickly:

• Derive \(f'(x)\) and \(f''(x)\) efficiently.
• Plot these functions over a desired range to visually analyze the behavior of the original function.
• Observe points of interest such as zero crossings, which may indicate relative extrema or inflection points.

CAS greatly enhances the ability to analyze functions and gain insights into their behavior without tedious calculations.

Inflection Points

Inflection points represent where a function changes its concavity. This means it's the point where a graph shifts from being concave up (shaped like a cup \((\cup)\)) to concave down (shaped like a cap \((\cap)\)), or vice versa. These points are crucial for understanding the function's overall shape and transition across its domain.
To find inflection points using derivatives:

• Use the second derivative \(f''(x)\).
• An inflection point happens where \(f''(x)\) changes sign.

Inflection points provide valuable information about the curvature and infer the points where the rate of change of the slope is exchanged.