Problem 46 - Calculus Early Transcendentals

Question

If \(f\) is a periodic function, then the locations of all absolute extrema on the interval \((-\infty,+\infty)\) can be obtained by finding the locations of the absolute extrema for one period and using the periodicity to locate the rest. Use this idea in these exercises to find the absolute maximum and minimum values of the function, and state the \(x\) -values at which they occur. \(f(x)=3 \cos \frac{x}{3}+2 \cos \frac{x}{2}\)

Step-by-Step Solution

Answer

Maximum and minimum values are periodic with a period of \(12\pi\).

Step 1: Determine the Period of the Function

To determine the period of the function, find the least common multiple (LCM) of the periods of the individual cosine terms. The first term, \(3 \cos \frac{x}{3}\), has a period of \(6\pi\), and the second term, \(2 \cos \frac{x}{2}\), has a period of \(4\pi\). The LCM of \(6\pi\) and \(4\pi\) is \(12\pi\). Thus, the period of the function \(f(x)\) is \(12\pi\).

Step 2: Find Critical Points in One Period

Focus on one period, such as \([0, 12\pi]\), to find the critical points. First, calculate the derivative \(f'(x)\) by differentiating each term: \(-\sin \frac{x}{3} \times \frac{1}{3}\) and \(-\sin \frac{x}{2} \times \frac{1}{2}\). So, \(f'(x) = -\sin \frac{x}{3} \cdot \frac{1}{3} -\sin \frac{x}{2} \cdot \frac{1}{2}\). Set \(f'(x) = 0\) to find critical points.

Step 3: Solve the Equation for Critical Points

Solving \(-\frac{1}{3} \sin \frac{x}{3} - \frac{1}{2} \sin \frac{x}{2} = 0\) involves trigonometric identities and possibly numerical methods. The solution must be found within the interval \([0, 12\pi]\). Identify any critical points as potential locations for extrema.

Step 4: Evaluate Function at Critical Points

With critical points identified, evaluate \(f(x)\) at each critical point within the period \([0, 12\pi]\) as well as at the endpoints \(0\) and \(12\pi\). Compare these values to locate the absolute maximum and minimum in this interval.

Step 5: Use Periodicity to Determine Global Extrema

Once the extrema within \([0, 12\pi]\) are known, utilize the periodicity. Since the function is periodic, the absolute extrema recur every \(12\pi\) units along the x-axis. Describe these values across the interval \((-\infty, +\infty)\).

Key Concepts

Absolute Extrema

In mathematics, finding the absolute extrema of a function means identifying the highest and lowest values that the function can achieve. These values are called the absolute maximum and absolute minimum.
For periodic functions, which repeat their values at regular intervals, finding the absolute extrema can be more systematic. Instead of analyzing the entire infinite domain, focus on a single period of the function. Once these extrema are identified in one period, they will appear repeatedly throughout.
Generally, the absolute extrema of a periodic function on the whole real line can be found by:

• Finding extrema in one period
• Applying the function's periodicity. This means plotting the extrema in one period at regular intervals over the domain.

For example, if a function has a period of 12",

Critical Points

Critical points of a function are the values of x at which its derivative is zero or undefined. These points are essential in determining where the extrema (either maxima or minima) might occur.
For periodic functions, critical points are found within a single period. After finding these, researchers evaluate the function's value at these points and the endpoints of the interval to identify extrema.
Thanks to calculus, you can find these critical points by taking the derivative of the function and setting it to zero. This process will reveal crucial x-values where the function's slope changes from increasing to decreasing or vice-versa.

• The derivative is symbolized by \( f'(x) \).
• Solve \( f'(x) = 0 \) to find critical points.
• Evaluate function at these points to determine potential extrema.

Understanding the significance of critical points helps in finding the absolute extrema of periodic functions, especially within specified intervals.

Function Period

A function's period is the length of the smallest interval over which the function completes one full cycle of its behavior and starts to repeat. For trigonometric functions, periods are characterized by their regular oscillations.
To find the period of a composite trigonometric function, like a combination of cosine terms, you look for the least common multiple (LCM) of the individual periods.
Here's how it works:

• Identify periods of each trigonometric component.
• Use the LCM to determine the complete period of the composite function.
• This period tells you how often the function starts repeating itself completely.

For example, a function composed of terms with periods of \(6\pi\) and \(4\pi\) has an overall period of \(12\pi\), which is the LCM in this case. Understanding the function period is crucial for determining where extrema will recur on the entire real line.

Trigonometric Equations

Trigonometric equations are equations involving trigonometric functions like sine, cosine, and tangent. These equations frequently arise in problems concerning periodic functions as their fundamental property is their periodicity.
Solving a trigonometric equation often involves applying identities and algebraic manipulations to find solutions within a specific interval. For functions like \(f(x)=3 \cos \frac{x}{3} + 2 \cos \frac{x}{2}\), it requires handling terms with different angles and periods.
Here's a structured approach:

• Start by simplifying and restructuring the equation.
• Apply trigonometric identities if needed, like the Pythagorean identities.
• Focus on solutions within a single period or specified interval, extending the solutions to other periods using the function's periodicity.

Solving these equations helps in identifying critical points and further determining the extrema of the periodic functions, as seen in multiple mathematical applications.