Problem 58 - Precalculus Mathematics for Calculus

Question

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer rounded to two decimal places. $$y=x^{5}-5 x^{2}+6, \quad[-3,3] \text { by }[-5,10]$$

Step-by-Step Solution

Answer

Local maximum: (0, 6); Local minimum: (1.26, 1.57).

Step 1: Understand the Polynomial

The polynomial given is \( y = x^5 - 5x^2 + 6 \). This is a fifth degree polynomial, which indicates the graph may have up to 4 turning points and slight oscillations.

Step 2: Determine the Derivative

To find local extrema, we first need the derivative of the function. Calculate the derivative:\[ y' = \frac{d}{dx}(x^5 - 5x^2 + 6) = 5x^4 - 10x \].

Step 3: Find Critical Points

Set the derivative equal to zero to find critical points:\[ 5x^4 - 10x = 0 \].Factor out the common term:\[ 5x(x^3 - 2) = 0 \].So, the critical points are where:\[ x = 0 \] or \[ x^3 - 2 = 0 \].Solve \( x^3 = 2 \) for \( x \):\[ x = \sqrt[3]{2} \approx 1.26 \].

Step 4: Evaluate Critical Points

Evaluate the original polynomial at the critical points to find the corresponding \( y \)-values:For \( x = 0 \):\[ y = 0^5 - 5(0)^2 + 6 = 6 \].For \( x = 1.26 \):\[ y = (1.26)^5 - 5(1.26)^2 + 6 \approx 1.57 \].Verify by checking whether these are maxima or minima using the second derivative or by visual inspection of the graph.

Step 5: Graph the Polynomial

Using a graphing calculator or software, graph the polynomial \( y = x^5 - 5x^2 + 6 \) in the viewing rectangle \([-3, 3] \) by \([-5, 10]\). This visualization helps confirm the positions of local extrema found.

Step 6: Verify Local Extrema

Inspect the graph to confirm the local extrema. The critical point \( x = 0 \) results in a local maximum with \((0, 6)\). The point \( x \approx 1.26 \) results in a local minimum with coordinates approximately \((1.26, 1.57)\).

Key Concepts

Fifth Degree Polynomial

A fifth degree polynomial is an algebraic expression with the highest exponent of the variable being five. It takes the general form \( ax^5 + bx^4 + cx^3 + dx^2 + ex + f \), where \( a, b, c, d, e, \) and \( f \) are coefficients. These polynomials can be quite interesting because they feature complex behavior, including up to four turning points. For instance, the polynomial \( y = x^5 - 5x^2 + 6 \) is a fifth degree polynomial.
It can exhibit maxima, minima, or inflection points depending on its derivative's behavior. This rich behavior makes graphing essential to understand its shape and characteristics.
Understanding the degree of a polynomial is crucial because it informs us about the potential complexity of its graph, including the number of turning points or possible real roots. Such insights are essential when predicting how the polynomial behaves for different values of \( x \).

Local Extrema Calculation

Local extrema refer to points on a graph where the function takes on a local minimum or maximum value. These points occur where the graph changes direction, which is often at turning points.
To find local extrema, we need to evaluate the function at its critical points. Critical points occur where the derivative of the function is zero or undefined. For example, for the polynomial \( y = x^5 - 5x^2 + 6 \), after finding the derivative \( y' = 5x^4 - 10x \), we set it equal to zero to solve for critical points:

• Factor the derivative: \( 5x(x^3 - 2) = 0 \)
• Solve for \( x \): \( x = 0 \) or \( x = \sqrt[3]{2} \), approximately 1.26

Evaluating the original function at these critical points gives us the \( y \)-values, which represent the heights of the local extrema.

Derivative and Critical Points

The derivative of a function is a crucial tool for finding critical points, which are potential locations of local extrema. By taking the first derivative, we determine where the slope of the tangent to a function is zero, indicating possible maxima or minima.
For the polynomial \( y = x^5 - 5x^2 + 6 \), the first derivative is calculated as \( y' = 5x^4 - 10x \). When we set this derivative to zero, we find critical points:

• First, factor the derivative: \( 5x(x^3 - 2) = 0 \)
• Solve for \( x \): \( x = 0 \) and \( x = \sqrt[3]{2} \)

Once the critical points are identified, the next step is to determine the function values at these points, which are used to assess if these points are local maxima or minima. The significance of these points is highlighted by their role in shaping the graph's contour.

Graphing Calculator Usage

Graphing calculators or graphing software are invaluable tools for visualizing functions like polynomials. They enable us to see the behavior of a function over a specified range. For our polynomial \( y = x^5 - 5x^2 + 6 \), we can enter the equation into a graphing calculator to verify our calculations of local extrema and to observe the overall shape of the graph in the given window with x-values from -3 to 3 and y-values from -5 to 10.
Using a graphing calculator involves:

• Entering the polynomial equation into the graphing function
• Adjusting the viewing window to capture all important behavior
• Inspecting the graph to confirm calculated extrema positions and heights

This process can confirm calculations, such as our identified extrema at \((0, 6)\) and \((1.26, 1.57)\), as well as reveal insights into higher degree polynomials and their complex nature, thus supporting analytical work with visual evidence.