Problem 79 - Precalculus Mathematics for Calculus

Question

(a) Graph the function \(P(x)=(x-2)(x-4)(x-5)\) and determine how many local extrema it has. (b) If \(a

Step-by-Step Solution

Answer

The function has two local extrema, located at \(x = 4\) and \(x = 5\). Any cubic function with distinct roots like \((x-a)(x-b)(x-c)\) will have two local extrema.

Step 1: Identify the Function Type

The function given is \(P(x) = (x-2)(x-4)(x-5)\). It is a polynomial function of degree 3 (cubic function) because it is the product of three linear terms, \((x-2), (x-4), \) and \((x-5)\).

Step 2: Find the Derivative

To find the local extrema, we first need to find the derivative of \(P(x) = (x-2)(x-4)(x-5)\). Use the product rule to differentiate:\[P'(x) = \left( (x-4)(x-5) + (x-2)(x-5) + (x-2)(x-4) \right).\]

Step 3: Expand and Simplify the Derivative

Expand and simplify the terms in the derivative found in Step 2:\[P'(x) = (x-4)(x-5) + (x-2)(x-5) + (x-2)(x-4) = x^2 - 9x + 20.\]

Step 4: Find Critical Points

Set the derivative equal to zero to find critical points:\[x^2 - 9x + 20 = 0.\]Factor to get \((x-4)(x-5) = 0\), so the critical points are \(x = 4\) and \(x = 5\).

Step 5: Determine Local Extrema

Evaluate the second derivative to determine the nature of the critical points. The second derivative \(P''(x)\) gives the concavity:\[P''(x) = 2x - 9.\]Substitute \(x = 4\) and \(x = 5\) to see the concavity and confirm these as local extrema. With different concavity signs at \(x = 4\) and \(x = 5\), both are local extrema.

Step 6: Graph the Function

Plot the function \(P(x) = (x-2)(x-4)(x-5)\) using the critical points found. Ensure there are changes in sign around these points in the first derivative that affirm local maxima and minima.

Step 7: Analyze the General Case

Using the general function \(P(x) = (x-a)(x-b)(x-c)\) where \(a < b < c\), the derivative will have two roots, \(b\) and the midpoint of \(a\) and \(c\). Therefore, a cubic polynomial with distinct roots has exactly two local extrema by this property.

Key Concepts

Derivative Calculation

The derivative of a function helps us understand how the function is changing at any given point. For a cubic function like \(P(x) = (x-2)(x-4)(x-5)\), we use the product rule to find the first derivative, \(P'(x)\). This involves differentiating each term separately and summing the results:

• \((x-4)(x-5)\)
• \((x-2)(x-5)\)
• \((x-2)(x-4)\)

After applying the product rule and simplifying, you get \(P'(x) = x^2 - 9x + 20\).
This derivative expresses the rate of change of the function and is crucial to finding critical points, which can lead us to local maxima or minima.

Local Extrema

Local extrema refer to the peaks and troughs of a function, which are the local maximum and minimum values that a function can take within a given interval. These points occur where the derivative equals zero and can be further verified using the second derivative test.
For \(P(x) = (x-2)(x-4)(x-5)\), the critical points where \(P'(x) = 0\) are solved by factoring the derivative as \((x-4)(x-5) = 0\), giving us \(x = 4\) and \(x = 5\).
Using the second derivative \(P''(x) = 2x - 9\), we substitute these x-values. If \(P''(x) > 0\), it indicates a local minimum, while \(P''(x) < 0\) indicates a local maximum.

Critical Points

Critical points are locations on the graph where the rate of change (the derivative) is zero or undefined. These points are important because they potentially indicate locations of local extrema or inflection points.
For our function \(P(x)\), setting \(P'(x) = x^2 - 9x + 20\) equal to zero gives us the equation \((x-4)(x-5) = 0\). The solutions to this equation, \(x = 4\) and \(x = 5\), are critical points. These points are calculated straightforwardly by solving the factors after setting the derivative to zero.
Examining these further with the second derivative, we determine whether each point is a maximum or minimum.

Graphing Polynomials

Graphing a polynomial like a cubic function involves using information such as critical points, intercepts, and the behavior at infinity to sketch its shape. For \(P(x) = (x-2)(x-4)(x-5)\), we found critical points at \(x = 4\) and \(x = 5\). With this information, plot the polynomial step by step.

• Determine where the polynomial intersects the x-axis, which occurs at the roots: \(x = 2\), \(x = 4\), and \(x = 5\).
• Using the critical points, determine the changes in sign of the first derivative, which showcases where the function increases and decreases.
• The second derivative can be used to determine concavity; thus, aiding in sketching the curve accurately.

By analyzing these characteristics, the graph represents how the cubic polynomial behaves, which includes its local maxima and minima.