Problem 53 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The local maximum is at (-1, 5) and the local minimum is at (1, 1).

Step 1: Identify Critical Points

To find local extrema, we need to identify the critical points of the polynomial. First, find the derivative of the function: \[ y' = \frac{d}{dx}(3x^5 - 5x^3 + 3) = 15x^4 - 15x^2. \] Set the derivative equal to zero and solve for \(x\): \[ 15x^4 - 15x^2 = 0 \] Factoring gives: \[ 15x^2(x^2 - 1) = 0. \] Thus, the critical points are \(x = 0, 1, -1.\)

Step 2: Evaluate the Second Derivative

Determine the nature of each critical point by evaluating the second derivative to test for concavity:\[ y'' = \frac{d}{dx}(15x^4 - 15x^2) = 60x^3 - 30x. \]Evaluate the second derivative at each critical point:1. \( y''(0) = 0 \)2. \( y''(1) = 60(1)^3 - 30(1) = 30 \) (positive, hence local minimum)3. \( y''(-1) = 60(-1)^3 - 30(-1) = -30 \) (negative, hence local maximum)

Step 3: Calculate Coordinates of Extrema

Substitute the critical points back into the original function to find the coordinates of local extrema:1. \(y(0) = 3(0)^5 - 5(0)^3 + 3 = 3\) (point is (0, 3))2. \(y(1) = 3(1)^5 - 5(1)^3 + 3 = 1\) (local minimum at (1, 1))3. \(y(-1) = 3(-1)^5 - 5(-1)^3 + 3 = 5\) (local maximum at (-1, 5))

Step 4: Graph the Polynomial

Graph the polynomial \(y = 3x^5 - 5x^3 + 3\) over the specified viewing rectangle \([-3, 3]\) by \([-5, 10]\). Plot the local extrema points identified:- Local maximum at \((-1, 5)\)- Local minimum at \((1, 1)\)- Additional point \((0, 3)\)

Key Concepts

Critical Points

Critical points in a polynomial graph are where the derivative equals zero or is undefined. These points often signal where the graph's slope changes, typically leading to local maxima or minima. Finding critical points of a polynomial function involves derivative calculation and setting it to zero.
To find these points for the polynomial \(y = 3x^5 - 5x^3 + 3\), calculate its first derivative: \(y' = 15x^4 - 15x^2\). Set the derivative equal to zero:

• \(15x^4 - 15x^2 = 0\)
• Factor the equation as \(15x^2(x^2 - 1) = 0\)
• Solve for \(x\): resulting in critical points \(x = 0, 1, -1\)

Once these points are identified, you have potential locations for local extrema.

Local Extrema

Local extrema in the context of polynomial graphing are the highest or lowest points in a local neighborhood of the function. They appear as hills or valleys in the graph. To determine whether a critical point is a local maximum or minimum, you evaluate the original function at these points and use the second derivative test.
In this exercise, critical points occur at \(x = 0, 1, -1\). After calculating, you get:

• \(y(0) = 3\)
• \(y(1) = 1\) (local minimum)
• \(y(-1) = 5\) (local maximum)

These evaluations reveal that at \((1, 1)\), the graph reaches a local minimum, and at \((-1, 5)\), it has a local maximum.

Second Derivative

The second derivative provides insight into the concavity of the function at critical points, helping to verify whether these points are maxima or minima. By evaluating the second derivative, the nature of each critical point is defined by whether the second derivative is positive or negative.
Compute the second derivative of \(y = 3x^5 - 5x^3 + 3\): \(y'' = 60x^3 - 30x\). Evaluate it at the critical points:

• \(y''(0) = 0\)
• \(y''(1) = 30\) (positive, indicates a local minimum)
• \(y''(-1) = -30\) (negative, indicates a local maximum)

A positive second derivative at a critical point implies a local minimum because the graph is concave up, while a negative second derivative indicates a local maximum as the graph is concave down.

Polynomial Derivative

Understanding polynomial derivatives is crucial to graphing and analyzing functions. The first derivative of a polynomial function provides the slope of the tangent at any point on its graph, thus helping find critical points where the slope equals zero.
Given the function \(y = 3x^5 - 5x^3 + 3\), its derivative is calculated as \(y' = 15x^4 - 15x^2\). This derivative is pivotal for:

• Finding critical points by setting \(y' = 0\)
• Determining where the slope is zero or changes sign

By mastering the mechanics of polynomial derivatives, students gain the ability to deduce a lot about the graph's behavior, including increasing or decreasing intervals and locating points of inflection.