Problem 45 - Thomas Calculus

Question

Identify the coordinates of any local and absolute extreme points and inflection points. Graph the function. \begin{equation} y=\left|x^{2}-1\right| \end{equation}

Step-by-Step Solution

Answer

Local minima at \((1, 0)\) and \((-1, 0)\); an inflection point at \((0, 1)\).

Step 1: Understand the Function

We are given the function \( y = |x^2 - 1| \). This is an absolute value function, which means it will reflect any negative values of \( x^2 - 1 \) over the x-axis to make them positive.

Step 2: Analyze the Critical Points

The expression \( x^2 - 1 \) is zero at \( x = \pm 1 \), thus these are points where the function changes form. These points are critical for determining local and absolute extrema.

Step 3: Determine Local and Absolute Extrema

Evaluating \( y \) at \( x = 1 \) and \( x = -1 \) gives \( y = 0 \). Both these points are where the function changes direction, thus \((1, 0)\) and \((-1, 0)\) are local minima. For large \( |x| \), \( y \) increases without bound, so there are no absolute maxima.

Step 4: Identify Potential Inflection Points

An inflection point occurs where the concavity changes. Check the function \( x^2 - 1 \) around \( x = 0 \). At \( x = 0 \), \( y = 1 \). Since the curvature changes at this point from concave down to concave up, it is an inflection point.

Step 5: Graph the Function

Sketch the graph of the function. It is V-shaped (due to absolute value) with vertex minima at \( x = 1 \) and \( x = -1 \). The function decreases from infinity to 0 as \( x \) approaches \( -1 \) and \( 1 \). After that, it increases back to infinity from \( x = 1 \) onward. At \( x = 0 \), the graph has an inflection point.

Key Concepts

Extreme Points

Extreme points in a function include local maxima, local minima, absolute maxima, and absolute minima. For the absolute value function \( y = |x^2 - 1| \), local minimum points are points where the function value is less than its neighbors. Here, \( x = 1 \) and \( x = -1 \) are local minima since \( y = 0 \) at these points, and the curve dips down to these points from both directions.

Absolute maximum points are values where the function takes on the greatest value on its domain. In this function, however, because as \( |x| \) becomes large, \( y \) increases indefinitely, no absolute maximum exists.

Occasionally, you may need to check endpoints or use limits, but for this function, inspecting the behavior at noted points \( x = 1 \) and \( x = -1 \), we observe they truly identify the local minima uniquely as well as part of the V-shape structure of this function.

Inflection Points

An inflection point is where a curve changes concavity, moving from being concave up to concave down or vice versa. Within our function \( y = |x^2 - 1| \), potential inflection points occur where \( x^2 - 1 \) changes its concavity.

Take \( x = 0 \); here \( y(0) = 1 \). The function transitions from concave down to concave up, marking \( x=0 \) as an inflection point. Visualize it as how the curve appears naturally V-shaped and represents a softer bend at this origin spot. This soft transition in curve shape is crucial to comprehending inflection points.

• Check derivative behavior or graphical changes to affirm inflection changes.
• Inflection points don't necessarily have to correlate with extreme points.

Understanding inflection points aids in appreciating the overall flow and aesthetic of the function graph.

Absolute Value Function

The absolute value function is famous for transforming negative values to positive values, rendering a distinctive V or U shape on a graph. In this case, the function \( y = |x^2 - 1| \) alters any negative output from \( x^2 - 1 \) into a positive result. Effectively, for this function:

• If \( x^2 - 1 \) is positive, \( y = x^2 - 1 \).
• If \( x^2 - 1 \) is negative, \( y = -(x^2 - 1) \).

Knowing this guides understanding how absolute value functions retain v-shaped forms, mirroring negative parts of their input function across the x-axis. Such functions are often simple yet expressive in their transformations!

Graphing Functions

Graphing functions is an artful blend of mathematics and visual representation. For \( y = |x^2 - 1| \), envision a symmetrical V centered along the x-axis with its vertex dropping to zero at \( x = \pm 1 \). Start by plotting critical points, like resultant minima and noted inflection points.

To graph efficiently:

• Plot the vertex and critical points accurately.
• Observe symmetry, especially with absolute value functions.
• Illustrate the approach towards infinity with large \( |x| \). This shows graph behavior at extreme x-values.

By graphing, the function's V structure highlights extreme (local minimum) points at \( x = \pm 1 \), while ensuring an intuitive understanding of how the function behaves across different domains!