Problem 57 - Algebra and Trigonometry

Question

\(51-58\) Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer rounded to two decimal places. $$ y=3 x^{5}-5 x^{3}+3, \quad[-3,3] \text { by }[-5,10] $$

Step-by-Step Solution

Answer

Local maximum at (-1, 11), local minimum at (1, 1).

Step 1: Understand the Problem

We are given the polynomial \(y = 3x^5 - 5x^3 + 3\) and asked to graph it over the specified viewing rectangle \([-3, 3] \times [-5, 10]\). We will also find the local extrema (maximum and minimum points) and round them to two decimal places.

Step 2: Find the Derivative

To find the local extrema, we need the derivative of the polynomial. The derivative is given by: \[ y' = \frac{d}{dx}(3x^5 - 5x^3 + 3) = 15x^4 - 15x^2 \]. We'll use this to find critical points by setting \(y' = 0\).

Step 3: Solve for Critical Points

Set the derivative equal to zero and solve for \(x\): \[ 15x^4 - 15x^2 = 0 \]. Factor out common terms:\[ 15x^2(x^2 - 1) = 0 \]. This results in \(x^2 = 0\) and \(x^2 = 1\). Thus, the critical points are \(x = 0\), \(x = -1\), and \(x = 1\).

Step 4: Evaluate the Function at Critical Points

Substitute the critical points back into the original function to find their corresponding \(y\)-values:- For \(x = 0\): \[ y = 3(0)^5 - 5(0)^3 + 3 = 3 \]- For \(x = 1\): \[ y = 3(1)^5 - 5(1)^3 + 3 = 1 \]- For \(x = -1\): \[ y = 3(-1)^5 - 5(-1)^3 + 3 = 11 \].

Step 5: Determine Local Extrema

The local maximum and minimum values are identified by comparing the \(y\) values at the critical points:- Local maximum at \((-1, 11)\)- Local minimum at \((1, 1)\).Another critical point at \((0, 3)\) represents neither a maximum nor a minimum.

Step 6: Sketch the Graph

Plot the polynomial \(y = 3x^5 - 5x^3 + 3\) over the viewing rectangle \([-3, 3] \times [-5, 10]\). Indicate the points \((-1, 11)\), \((0, 3)\), and \((1, 1)\) on the graph. The polynomial exhibits behavior corresponding to extrema at these points.

Key Concepts

Local Extrema

When graphing a polynomial, local extrema refer to the highest or lowest points on specific segments of the graph. These include local maxima, where the graph reaches a high point, and local minima, where it reaches a low point in that specific neighborhood.
To identify these points, you evaluate the critical points—places where the derivative is zero or undefined. At these points, the function changes direction.
The local extrema can be determined by comparing the function values at each critical point.

• A local maximum occurs when a function value is higher than the values at both of its neighboring points.
• A local minimum occurs when a function value is lower than the values at both of its neighboring points.

In our example, the polynomial reaches a local maximum at the point (-1, 11) and a local minimum at (1, 1). These points were determined by comparing calculated values from the critical points.

Finding Derivatives

Finding derivatives is an essential process for analyzing the behavior of polynomial functions. Derivatives provide us with a function that describes the rate of change or slope of the original function.
In polynomial graphing, calculating the derivative is crucial for identifying critical points, which can indicate the location of local extrema.
When computing the derivative of a polynomial, you use simple rules:

• The derivative of a constant is zero.
• For a term written as a coefficient multiplied by a variable raised to a power, the derivative is found by multiplying the coefficient by the power, and then reducing the power by one.

In the given polynomial \(y = 3x^5 - 5x^3 + 3\), the derivative \(y'\) can be calculated as \(15x^4 - 15x^2\). This demonstrates the use of these rules to derive expressions indicative of the polynomial's slope at any point.

Critical Points

Critical points in polynomial graphing are where the derivative of the function equals zero or is undefined. These points are significant as they can indicate potential local maxima, minima, or points of inflection.
For a polynomial function like \(y = 3x^5 - 5x^3 + 3\), finding critical points involves setting its derivative equal to zero and solving: \(15x^4 - 15x^2 = 0\).
To solve, factor the expression: \(15x^2(x^2 - 1) = 0\). By breaking it down, you find possible solutions:

• \(x = 0\)
• \(x^2 = 1\), leading to \(x = 1\) and \(x = -1\)

These provide the critical points, \(-1\), \(0\), and \(1\). Identifying these points helps to determine the function's important behaviors, like where the graph turns or changes direction, indicating local extrema or other significant characteristics.