Question

The graph of \(y=f(x)\) depends on a parameter c. Using a CAS, investigate how the extremum and inflection points depend on the value of \(c .\) Identify the values of \(c\) at which the basic shape of the curve changes. $$ f(x)=\frac{c x}{4+(c x)^{2}} $$

Short answer

The shape changes as values of \(c\) impact extrema at \(x=\pm\frac{2}{c}\) and inflection point's behavior through complex analysis.

Step-by-step solution

Understand the Function and Its Derivatives

The given function is \(f(x) = \frac{cx}{4+(cx)^2}\). To investigate its extrema and inflection points, we need to find its first and second derivatives. Finding these derivatives gives us insights into the behavior of the function concerning changes in the parameter \(c\).

Find the First Derivative for Extrema

The first derivative \(f'(x)\) is required to find the extrema of the function. Using the quotient rule, compute \(f'(x)\):\[f'(x) = \frac{d}{dx} \left( \frac{cx}{4+(cx)^2} \right) = \frac{c(4 - c^2x^2)}{(4 + c^2x^2)^2}\].Set \(f'(x) = 0\) to find critical points, which will help in determining the extremum points.

Solve for Critical Points

To find critical points, solve \(\frac{c(4 - c^2x^2)}{(4 + c^2x^2)^2} = 0\). This equation simplifies to:\[4 - c^2x^2 = 0\] \[x^2 = \frac{4}{c^2}\] Thus, the critical points are \(x = \pm\frac{2}{c}\). These points give us potential extrema depending on the values of \(c\).

Find the Second Derivative for Inflection Points

The second derivative \(f''(x)\) tells us about the concavity of the function and helps locate inflection points. Compute \(f''(x)\) using the quotient rule applied to \(f'(x)\):\[f''(x) = \frac{d}{dx} \left( \frac{c(4 - c^2x^2)}{(4 + c^2x^2)^2} \right)\].This process is complex and computational software (CAS) should be used.

Find Inflection Points

Set \(f''(x) = 0\) to determine points of inflection. The CAS solution will depend on solving the expression obtained from differentiating \(f''(x)\).

Analyze Changes in Basic Shape

The shapes of curves change at values of \(c\) where characteristics such as the number or nature of critical points change. Typically, this will be where we have a shift from real to complex roots or where extrema and inflection point behavior alters structurally. Solving the expressions for the derivatives through CAS, identify for which \(c\) the sign of the second derivative changes at critical points indicating shape change.

Key concepts

Parameter Analysis

Parameter analysis in calculus involves understanding how changes in a parameter, such as \(c\) in the function \(f(x) = \frac{cx}{4+(cx)^2}\), affect the function's behavior, specifically its graphical representation. Entirely determining these effects involves:

• Analyzing the derivatives using the parameter.
• Exploring how changes in \(c\) can affect critical points and inflection points.
• Observing generalized patterns as \(c\) varies.

For instance, differing values of \(c\) alter where and if extrema and inflection points appear, thereby modifying the curve's basic shape. This critical investigation reveals insights into the multitude of behaviors a parameter-controlled function can exhibit.

Extremum Points

Extremum points, comprising of maxima and minima, occur at values of \(x\) where the derivative \(f'(x)\) is zero or undefined. For the function \(f(x) = \frac{cx}{4+(cx)^2}\), the first derivative \(f'(x)\) is computed as: \[f'(x) = \frac{c(4 - c^2x^2)}{(4 + c^2x^2)^2}\].To determine extremum points, one solves \(f'(x) = 0\), which simplifies to \[4 - c^2x^2 = 0\] yielding critical points \(x = \pm \frac{2}{c}\).These critical points indicate potential extremum points. To specify if these are maxima or minima, one can use the second derivative test or analyze the sign changes in \(f'(x)\) around these points.

Inflection Points

Inflection points are found where the second derivative \(f''(x)\) changes its sign, implying a change in concavity. In our function \(f(x) = \frac{cx}{4+(cx)^2}\), the analysis of inflection requires solving \(f''(x) = 0\). While the exact formulation for \(f''(x)\) is complex, leveraging a Computer Algebra System (CAS) aids in deriving and solving this equation.By solving \(f''(x) = 0\), one identifies inflection points, which are valuable for understanding how \(f(x)\) bends or doesn't bend at certain \(x\) values depending on the chosen \(c\). These points play a crucial role in the curve's shape alterations over the parameter \(c\).

Graphical Behavior

The graphical behavior of \(f(x) = \frac{cx}{4+(cx)^2}\) is influenced by the derived critical and inflection points. Changing \(c\) alters these loci, hence modifying the curve's profile. To visualize the effects, consider:

• For small \(c\), the critical points \(x = \pm \frac{2}{c}\) are far apart, resulting in a stretched graph.
• For large \(c\), these points come closer, possibly merging or disappearing, indicating altered behavior.
• The appearance or loss of inflections points contributes significantly to shifts in concavity and overall shape.

Thus, the graphical behavior of the function encapsulates the dynamic nature of how \(c\) influences geometry.

Critical Points

Critical points are points on the graph where the derivative \(f'(x)\) is zero or undefined, leading to potential extremum (maxima or minima) and possibly influencing inflection. In \(f(x) = \frac{cx}{4+(cx)^2}\), these are obtained by finding the roots of:\[4 - c^2x^2 = 0\], leading to \(x = \pm \frac{2}{c}\).This calculation shows where critical points lie, depending on \(c\). It represents locations where the trend shifts in the slope of the function, majorly affecting curve behavior and helping in identifying extremum and their nature based on further analysis, like concavity from \(f''(x)\). Critical points act as a compass directing how \(f(x)\) changes its path and narrative visually.