Problem 66 - Algebra and Trigonometry

Question

\(59-68\) Graph the polynomial and determine how many local maxima and minima it has. $$ y=\left(x^{2}-2\right)^{3} $$

Step-by-Step Solution

Answer

The polynomial has 1 local minimum and 0 local maxima.

Step 1: Understand the Polynomial

The polynomial given is \(y = (x^2 - 2)^3\). It is a cubic function, expressed in terms of a quadratic component \((x^2 - 2)\). This structure will help us understand the general shape of the graph.

Step 2: Find the Derivative

To find the local maxima and minima, we need to take the derivative of the function. Let \(u = x^2 - 2\). Hence, \(y = u^3\) and \(y' = 3u^2 \cdot u'\). Calculating \(u'\), we get \(2x\). Therefore, \(y' = 3(x^2 - 2)^2 \cdot 2x = 6x(x^2 - 2)^2\).

Step 3: Find Critical Points

Set \(y' = 6x(x^2 - 2)^2 = 0\). The derivative is zero when \(x = 0\) or \((x^2 - 2) = 0\). Solving \((x^2 - 2) = 0\), we get \(x = \pm\sqrt{2}\). Thus, the critical points are \(x = 0, \sqrt{2}, -\sqrt{2}\).

Step 4: Determine Nature of Critical Points

Use the second derivative test to determine whether each critical point is a local maximum or minimum. Calculate \(y''\) by differentiating \(y' = 6x(x^2-2)^2\) again. This results in a complex expression. For simplicity, evaluate the sign of \(y''\) at points between critical points and around them.

Step 5: Identify Local Maxima and Minima

Evaluate the second derivative at the critical points: \(x = 0\), \(y''(0)\) indicates a change in concavity, typically indicating local extremum points. Test intervals around the critical points to confirm the nature; \(x = \pm\sqrt{2}\) are found to be inflection points due to symmetry and changes in concavity.

Step 6: Graph the Polynomial

Graph \(y = (x^2 - 2)^3\). Identify local minima and maxima visually. The function's graph shows a local minima at \(x = 0\) and inflection points at \(x = \pm\sqrt{2}\).

Step 7: Count Local Maxima and Minima

Since \(x = 0\) is a local minima and \(x = \pm\sqrt{2}\) are inflection points without maxima, the function has 1 local minima and 0 local maxima.

Key Concepts

Local Maxima

In the context of polynomial graphing, local maxima are points on a graph where the function reaches a peak relative to its immediate surroundings. It's higher than any nearby points. To locate local maxima precisely, we need to explore the concept of derivatives. The basic idea is that at a local maximum, the slope of the tangent line is zero. This means the first derivative of the function at that point is zero. However, not every point where a derivative is zero is a local maximum. Some could be local minima or even inflection points. To confirm a point as a local maximum, use the second derivative test. If the second derivative is negative at a critical point, it indicates a concave down and confirms a local maximum.

Local Minima

Local minima are points on a graph where the function reaches its lowest value in the vicinity. It's lower than any nearby points. Much like with maxima, finding local minima involves solving when the derivative is zero to find critical points. Once critical points are located, apply the second derivative test to determine if it's a minimum. A positive second derivative at a critical point indicates concave up and confirms a local minimum. In our example, at the point \(x = 0\), the second derivative test signifies a local minimum as the concavity indicates the function curves upwards at this position.

Critical Points

A critical point is a point on the graph of a function where the derivative is zero or undefined. These points are essential as they highlight where potential extrema (minimum or maximum values) may occur. To identify these points, we set the first derivative equal to zero and solve for the variable. In our polynomial example, the critical points are found at \(x = 0\) and \(x = \pm\sqrt{2}\). These are potential points where the function may reach a local maximum, local minimum, or an inflection point which signifies a change in concavity.

Derivative

The derivative of a function is a fundamental concept in calculus that represents the rate of change or the slope of the function. In polynomial graphing, derivatives help us understand how the function behaves, particularly where it increases or decreases. The first derivative \(f'(x)\) is used to locate critical points by setting it equal to zero. This represents places where the slope of the tangent to the function is horizontal, and thus potential extrema are located. In our example, the first derivative \(y' = 6x(x^2 - 2)^2\) was crucial for identifying critical points \(x = 0\) and \(x = \pm\sqrt{2}\).

Second Derivative Test

The second derivative test is a method used to determine whether a critical point is a local minimum, local maximum, or neither. By evaluating the sign of the second derivative at critical points, we can infer the concavity of the function. - If \(y'' > 0\) at a critical point, the function is concave up, indicating a local minimum.- If \(y'' < 0\) at a critical point, the function is concave down, indicating a local maximum.- If \(y'' = 0\), it typically confirms an inflection point or requires further analysis.In the polynomial \(y = (x^2 - 2)^3\), using the second derivative test allowed us to verify that \(x = 0\) is a local minimum, while \(x = \pm\sqrt{2}\) are inflection points with no extremum.