Problem 54 - 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. $$ y=2 x^{3}-3 x^{2}-12 x-32, \quad[-5,5] \text { by }[-60,30] $$

Step-by-Step Solution

Answer

Local maximum at (-1, -39) and local minimum at (2, -52).

Step 1: Find the Derivative

To identify local extrema, we first need the derivative of the function. Given the polynomial function \( y = 2x^3 - 3x^2 - 12x - 32 \), the derivative is calculated as follows:\[ \frac{dy}{dx} = 6x^2 - 6x - 12 \]

Step 2: Set the Derivative to Zero

To find critical points, we set the derivative equal to zero:\[ 6x^2 - 6x - 12 = 0 \]

Step 3: Solve for x

To solve the quadratic equation \( 6x^2 - 6x - 12 = 0 \), we can simplify it:\[ x^2 - x - 2 = 0 \]Using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a=1, b=-1, c=-2 \), we get:\[ x = \frac{1 \pm \sqrt{1 + 8}}{2} = \frac{1 \pm 3}{2} \], yielding roots \( x = 2 \) and \( x = -1 \).

Step 4: Determine the Y-Values at Critical Points

Calculate the function value at the critical points found. Substitute \( x = 2 \) and \( x = -1 \) back into the original function:- \( f(2) = 2(2)^3 - 3(2)^2 - 12(2) - 32 = -52 \)- \( f(-1) = 2(-1)^3 - 3(-1)^2 - 12(-1) - 32 = -39 \)

Step 5: Identify and Classify Extrema

The critical points are (-1, -39) and (2, -52). To classify whether these points are local maxima or minima, we can perform sign tests around these points or use the second derivative test. Here, just by observation of the function's behavior (from its standard cubic shape and sign changes), \(-1\) is a local maximum and \(2\) is a local minimum.

Step 6: Graph and Confirm

Using the viewing window \([-5, 5] \) by \([-60, 30] \), plot the function \( y = 2x^3 - 3x^2 - 12x - 32 \) using graphing technology or software. Verify that the points \((-1, -39)\) and \((2, -52)\) match the observations, confirming them as local extrema.

Key Concepts

Local Extrema

In polynomial graphing, local extrema refer to the highest or lowest points in a specific region of a graph. These are where the function switches directions - either from increasing to decreasing or vice versa. Local extrema help us understand the behavior of a function over its domain.

To identify local extrema:

• Find critical points where the derivative is zero or undefined.
• Analyze the function behavior around these points.
• Identify whether these points are local maxima or minima based on function direction changes.

For example, in the function given, local extrema were identified at points (-1, -39) and (2, -52). By understanding these, students can ascertain important graph characteristics and confirm these through plotting.

Derivative Calculus

Derivative calculus is a powerful tool for analyzing functions, particularly when graphing polynomials. The derivative of a function represents the function's rate of change at any given point. This concept is crucial for identifying areas where the function's slope is zero, indicating potential local maxima or minima.

To find the derivative:

• For a polynomial, apply power rules: bring the power down and reduce the exponent by one.
• For the given function, the derivative is found as: \[ y' = 6x^2 - 6x - 12 \]

Calculating this derivative allows students to then set it to zero to find critical points, which is the next step in analyzing the polynomial's behavior.

Quadratic Formula Solving

The quadratic formula is a method for solving equations of the form \[ ax^2 + bx + c = 0 \] This formula, \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] is versatile for finding the roots of quadratic equations that arise in polynomial analysis.

In the exercise, the equation \[ 6x^2 - 6x - 12 = 0 \] was simplified to \[ x^2 - x - 2 = 0 \], where the quadratic formula was applied:

• Set coefficients as: \( a = 1 \), \( b = -1 \), \( c = -2 \)
• Compute roots: \( x = \frac{1 \pm \sqrt{1 + 8}}{2} = \frac{1 \pm 3}{2} \), giving roots \( x = 2 \) and \( x = -1 \).

Understanding this process is essential for navigating through more complex polynomial equations and identifying meaningful contributions from critical points.

Critical Points Identification

Critical points are where the derivative of a function equals zero or is undefined. These points hint where local extrema might reside, crucial for understanding polynomial graph shape. Identifying them involves finding where the derivative ceases to have a slope.

For the given function, the process involved:

• Calculating the derivative \[ 6x^2 - 6x - 12 \]
• Setting the derivative equation to zero: \[ 6x^2 - 6x - 12 = 0 \]
• Solving for \( x \) using the quadratic formula: roots found at \( x = -1 \) and \( x = 2 \)

Locating these critical points provides insights into where the function's graph might exhibit peaks or valleys. This analysis is validated further by computing the function values at these critical points to explicitly identify local maxima and minima, reinforcing calculus understanding.