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. $$ f(x)=x^{2} \sqrt{x}^{2}-c^{2} $$

Short answer

The extremum doesn't change with \(c\); the shape remains cubic, only vertically shifted by \(c^2\).

Step-by-step solution

Rewrite the Function

Start by rewriting the function for clarity. Given is \( f(x) = x^2 \sqrt{x}^2 - c^2 \). This can be simplified. Notice \( \sqrt{x}^2 = x \) for all real numbers. Thus, the function simplifies to \( f(x) = x^3 - c^2 \).

Find the First and Second Derivatives

To find the extremum and inflection points, differentiate \( f(x) = x^3 - c^2 \) with respect to \(x\). The first derivative is \( f'(x) = 3x^2 \). The second derivative is \( f''(x) = 6x \).

Determine Extremum Points

Set the first derivative equal to zero to find critical points: \( 3x^2 = 0 \). Solving this, we find \( x = 0 \) is a critical point. Evaluate \( f''(x) = 6x \) at this point; \( f''(0) = 0 \), indicating that \(x = 0\) is a potential inflection rather than extremum point.

Determine Inflection Points

Set the second derivative equal to zero: \( 6x = 0 \). Solving this, we find \( x = 0 \). As \( f''(x) \) changes sign at \( x = 0 \), this is indeed an inflection point.

Analyze Changes in \(c\) for Curve Shape

The function depends linearly on \(c^2\) as \( f(x) = x^3 - c^2 \). As \(c^2\) changes, it affects the vertical shift of the graph. The basic shape remains a cubic curve centered around the origin, unless \(c\) drastically approaches zero.

Conclusion on Values of \(c\)

Only the vertical shift of the graph changes due to \(c^2\). For significant changes in shape, \(c\) cannot be defined merely by its numeric value but affects particular real-world parameters attached to \(c\).

Key concepts

Extremum Points

In calculus, extremum points include both minimum and maximum points of a function. For a function to have an extremum at a certain point, the first derivative at that point must be zero.

To find extremum points, we first compute the derivative of the function and find its critical points. In the given cubic function \( f(x) = x^3 - c^2 \), we differentiate to get the first derivative as \( f'(x) = 3x^2 \). By setting this equal to zero, \( 3x^2 = 0 \), we find the critical point at \( x = 0 \).

This means there is no typical maximum or minimum since the second derivative test, \( f''(0) = 0 \), indicates a point of inflection rather than a clear extremum. At points where the second derivative is zero, further analysis is needed to conclude if it’s indeed an extremum.

Inflection Points

Inflection points are where the curve changes its concavity. These occur where the second derivative is zero, and a check shows a change in concavity.

For our cubic function, \( f(x) = x^3 - c^2 \), the second derivative is \( f''(x) = 6x \). Setting this equal to zero, \( 6x = 0 \), we find that \( x = 0 \) is a point where concavity changes.

This point is indeed an inflection point because the sign of \( f''(x) \) changes from negative to positive as x crosses zero. This fundamental change in direction at \( x = 0 \) demonstrates the concept and provides an understanding of how the curve behaves.

Graph Analysis

Graph analysis involves understanding the behavior and the shape of a function's graph. For a cubic function like \( f(x) = x^3 - c^2 \), the largest impact on shape is seen in how its inflection point and critical points change.

At \( x=0 \), the curve changes concavity, indicating an inflection point, which shows that the graph shifts in direction but remains smooth. The graph doesn't exhibit peaks or valleys at this point since there are no clear extremum points.

The overall graph retains the classic cubic shape, with one inflection that shifts vertically according to the parameter \( c^2 \). This insight helps students understand how shifting a parameter affects the curve's placement rather than its shape.

Cubic Function

Cubic functions are polynomial functions of degree three. They can exhibit a variety of shapes due to their higher degree, usually involving smooth, curved transitions between positive and negative infinite.

For cubic functions like \( f(x) = x^3 - c^2 \), the general shape varies from other polynomials due to the presence of inflection points and lack of a fixed symmetry. Cubic functions typically have one inflection point, as seen here at \( x = 0 \).

The graph is sensitive to parameter changes, like \( c^2 \), which affects its vertical position but not its shape. Understanding the nature of cubic functions is key in various applications, from modeling physical phenomena to economic predictions.

Parameter Investigation

Investigating parameters involves examining how changing a value in a function affects the graph. For \( f(x) = x^3 - c^2 \), the parameter \( c^2 \) is crucial in shifting the graph vertically.

Changing \( c \) doesn't affect the function's shape but alters its vertical position on the coordinate plane. This vertical shift can impact real-world phenomena the function models, hence understanding the effect of \( c \) is important.

As \( c \) approaches zero, the function's shift lessens, causing the graph to center more closely around the origin, while large values of \( c \) move it further along the vertical axis. Understanding parameter influence can deeply enhance comprehension of functional behavior and applications.