Problem 44 - A Textbook of Differential Calculus

Question

Statement I The maximum value of \(\sin \sqrt{2} x+\sin a x\) cannot be 2 . (where \(a\) is positive rational number) Statement II \(\frac{\sqrt{2}}{\alpha}\) is irrational.

Step-by-Step Solution

Answer

Statement II correctly explains why the maximum cannot be 2; \( \frac{\sqrt{2}}{a} \) is irrational.

Step 1: Analyze Statement I

We need to understand what it means for the maximum value of \( \sin \sqrt{2}x + \sin ax \) to be 2. The maximum value of a sine function is 1, so individually, both \( \sin \sqrt{2}x \) and \( \sin ax \) can achieve 1, but not simultaneously if they have different periods unless they are constructed carefully. Thus, if the sum is 2, both must achieve 1 at the same time, which requires that the functions synchronize their periods correctly.

Step 2: Explore Synchronization of Sine Functions

For two sine functions of different arguments to reach their maximum value of 1 simultaneously, their periods (cycles) must be synchronized at some integer multiple. Given \( \sin \sqrt{2}x \) and \( \sin ax \), synchronization implies a relation such as \( \frac{\sqrt{2}}{a} \) must be a rational number so that both functions can simultaneously achieve the maximum value.

Step 3: Evaluate Statement II

This statement says that \( \frac{\sqrt{2}}{\alpha} \) is irrational, which suggests that any divisor \(a\) of this expression must result in an irrational number because \( \sqrt{2} \) is irrational itself and non-rational numbers divided by rational numbers remain irrational.

Step 4: Connect Statements I and II

Statement II implies there are no rational values of \(a\) such that \( \frac{\sqrt{2}}{a} \) is rational. Therefore, Statement II disallows synchronization of the two sine functions because any rational \(a\) results in an irrational value. As a result, \( \sin \sqrt{2}x + \sin ax \) cannot both reach 1 simultaneously, which makes it impossible for the sum to reach 2.

Key Concepts

Sine Function Synchronization

Synchronized periods are key when you want sine functions to achieve their maximum value at the same time. Normally, the sine function, \( \sin(x) \), oscillates between -1 and 1. In this exercise, we face the challenge of the expressions \( \sin \sqrt{2}x \) and \( \sin ax \). Each of these sine functions can individually reach a maximum of 1.
However, their achieving 1 simultaneously requires synchronization. This feat can only occur if their periods align perfectly at some point. The period of a sine function \( y = \sin(bx) \) is given by \( \frac{2\pi}{b} \). For the functions to maximize together, their periods must be a common multiple. This means:

• \( \frac{\sqrt{2}}{a} \) needs to be rational.
• A rational number ensures that an integer multiple exists for periods to line up.
• If it remains irrational, synchronization fails.

In practical terms, to make both functions peak simultaneously at 1 is essentially impossible without this synchronization.

Irrational Numbers

Numbers like \( \sqrt{2} \) are termed irrational because they cannot be expressed as a simple fraction. Unlike rational numbers, they have non-terminating and non-repeating decimal expansions.
In this problem, we are dealing with the fraction \( \frac{\sqrt{2}}{\alpha} \) and its properties.

• For any rational number \( \alpha \), dividing by \( \alpha \) preserves the irrational nature of \( \sqrt{2} \).
• This is because the characteristics that define an irrational number are inherently preserved.

Thus, the statement that \( \frac{\sqrt{2}}{\alpha} \) remains irrational reinforces why synchronization in the earlier section fails. When you try to force a rational divisor, it conflicts with the core nature of \( \sqrt{2} \).

Maximum Value of Functions

When dealing with the maximum of a sine function like \( \sin(x) \), we know:

• The maximum value is uniformly 1.
• The cycle of maximum values is described by its period.

In our specific case with \( \sin \sqrt{2}x + \sin ax \), the concern lies in reaching the sum of these maxima, which reads as 2.
Reaching this sum simultaneously implies perfect timing; both functions reach their peak together. Highlight the fact that if they are unsynchronized, they never peak together. This limits their combined maximum value, explaining the impossibility of achieving 2.

Periodicity of Functions

The periodicity of a function, like sine, plays an integral role in its behavior. The sine waves \( \sin \sqrt{2}x \) and \( \sin ax \) each have distinct periods, crucial for our discussion.

• \( \sin(bx) \) has a period of \( \frac{2\pi}{b} \).
• Two functions synchronize perfectly only if their periods share a common multiple.

Given \( \frac{\sqrt{2}}{a} \) must be rational for synchronization, and given its inherent irrational nature, we learn:
- The functions remain "out of step," never aligning their peaks together.
Understanding this periodicity points directly to how the functions exhibit their behavior and limit their interactions. The mismatch in periodicity, driven by irrational nature, hints at why our attempt to simultaneously peak as 2 is not realized.