Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(f(x, y)=\cos x+\cos y+\cos (x+y)\); $$ 0

Step-by-Step Solution

Answer

The maximum value of \( f(x, y) \) is 3 as \( x, y \to 0 \).

Step 1: Understand the Function

The given function is a sum of cosine terms: \( f(x, y) = \cos x + \cos y + \cos (x+y) \). This is a trigonometric expression that depends on two variables, \( x \) and \( y \). Each cosine function has a maximum value of 1, as it ranges from -1 to 1.

Step 2: Analyze the Domain

The problem specifies the domain \( 0 < x < \frac{\pi}{2} \) and \( 0 < y < \frac{\pi}{2} \). Within this domain, \( \cos x \), \( \cos y \), and \( \cos(x+y) \) are all positive since cosine is positive in the first quadrant.

Step 3: Consider Values Close to Zero

As both \( x \) and \( y \) approach 0, \( \cos x \to 1 \) and \( \cos y \to 1 \), leading to \( \cos(x+y) \to 1 \) as well. At this point, the function \( f(x, y) \) approaches its theoretical maximum value of 3.

Step 4: Verify the Boundary Behavior

At the boundaries, where \( x \to \frac{\pi}{2} \) or \( y \to \frac{\pi}{2} \), the cosine of these angles becomes 0 because \( \cos(\frac{\pi}{2}) = 0 \). Likewise, as either \( x+y \to \frac{\pi}{2} \), \( \cos(x+y)\) will also approach 0. Hence \( f(x, y) \) decreases as we move towards these boundaries.

Step 5: Conclusion about Maximum Value

Based on these observations, the function \( f(x, y) \) is maximized as \( x \) and \( y \) approach 0. The value of the function at this maximum point is close to 3, given \( f(x, y) = 1 + 1 + 1 = 3 \).

Key Concepts

Trigonometric Identities

Trigonometric identities are crucial tools in the study of mathematics, especially when dealing with functions like the one in this exercise. These identities are equations that involve trigonometric functions and are true for any value of the variable within their domain. In this problem, the function is \( f(x, y) = \cos x + \cos y + \cos (x+y) \). Here are some important trigonometric identities that might help in understanding such expressions:

• Pythagorean Identities: \( \sin^2 x + \cos^2 x = 1 \).
• Angle Sum Identities: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \) can be used to rewrite cosine of sums.
• Double Angle Identity: \( \cos(2a) = \cos^2 a - \sin^2 a \), useful for simplifying certain expressions.

Understanding these fundamental relationships helps in not only simplifying expressions but also in finding the boundaries and maximum values of trigonometric functions within given domains.

First Quadrant Trigonometric Values

The first quadrant of the unit circle, where both \( x \) and \( y \) lie in this problem, is an essential region for evaluating trigonometric values. In the first quadrant, angles range from 0 to \( \frac{\pi}{2} \), and all trigonometric values are positive. Here's a quick guide on what happens in this quadrant:

• Cosine Values: \( \cos x \) and \( \cos y \) reach their maximum at 1 when the angles are near 0.
• Sine Values: While not immediately involved in the given function, note that \( \sin x \) and \( \sin y \) start from 0 and rise towards 1.

Since the function involves cosine, which remains positive here, and cosines tend to decrease as angles increase from 0 to \( \frac{\pi}{2} \), it's easy to see why the function's maximum occurs near the origin. Understanding the behavior of these values is crucial to solving and simplifying trigonometric functions.

Boundary Behavior in Calculus

Boundary behavior in calculus concerns examining and understanding how functions behave as inputs approach the edges of their allowed range. In this exercise, it is vital to consider what happens as \( x \) and \( y \) approach either end of their domain, which are within the first quadrant:

• As \( x \to 0 \) or \( y \to 0 \), cosine functions approach their maximum, leading to the entire expression potentially summing up to 3.
• Conversely, as \( x \to \frac{\pi}{2} \) or \( y \to \frac{\pi}{2} \), the cosine terms trend towards zero.

By analyzing boundary behavior, you can conclude that the function value drops as you move towards the upper boundaries of \( x \) and \( y \). This drop is because the cosine function approaches zero near \( \frac{\pi}{2} \). Checking these boundaries supports finding the maximum value point efficiently, ensuring you understand both the function's range and its extreme values.