the period of the function cos root 2x + cos 2x is. Qno 2 which of the following is true for. a real valued function y = f(x) dwfined on [-a,a]

To determine the period of the function \( \cos(\sqrt{2}x) + \cos(2x) \), we first need to analyze the periods of each individual cosine function. The period of a cosine function \( \cos(kx) \) is given by the formula \( \frac{2\pi}{|k|} \).

Finding the Periods of Each Component

Let's break it down:

• The first term is \( \cos(\sqrt{2}x) \). Here, \( k = \sqrt{2} \). Therefore, the period \( T_1 \) is:
• T₁ = \frac{2\pi}{\sqrt{2}} = \sqrt{2}\pi
• The second term is \( \cos(2x) \). In this case, \( k = 2 \). Thus, the period \( T_2 \) is:
• T₂ = \frac{2\pi}{2} = \pi

Finding the Least Common Multiple

To find the overall period of the combined function \( \cos(\sqrt{2}x) + \cos(2x) \), we need to find the least common multiple (LCM) of the two periods \( T_1 \) and \( T_2 \). The LCM of two periods will give us the smallest interval over which both functions complete an integer number of cycles.

We have:

• T₁ = \sqrt{2}\pi
• T₂ = \pi

To find the LCM, we can express \( T_1 \) in terms of \( T_2 \):

• Since \( \sqrt{2} \) is approximately 1.414, we can see that \( T_1 \) is not a simple multiple of \( T_2 \). Thus, we need to find a common multiple.

The LCM can be calculated as:

• LCM(\sqrt{2}\pi, \pi) = \sqrt{2}\pi

However, since \( \sqrt{2} \) is irrational, the LCM in this case is actually infinite, meaning the function \( \cos(\sqrt{2}x) + \cos(2x) \) does not have a finite period. It will repeat its values but not in a regular, finite interval.

Analyzing the Real-Valued Function

Now, regarding your second question about the real-valued function \( y = f(x) \) defined on the interval \([-a, a]\), we can consider some properties that are often true for such functions:

• If \( f(x) \) is an even function, then \( f(-x) = f(x) \) for all \( x \) in the domain. This symmetry about the y-axis is a key characteristic.
• If \( f(x) \) is an odd function, then \( f(-x) = -f(x) \). This means the function is symmetric about the origin.
• For any real-valued function defined on a symmetric interval, the behavior of the function at positive and negative values can provide insights into its overall shape and properties.

In summary, the function \( \cos(\sqrt{2}x) + \cos(2x) \) does not have a finite period due to the irrational nature of \( \sqrt{2} \), and for a real-valued function defined on \([-a, a]\), properties such as evenness or oddness can significantly influence its characteristics. If you have further questions or need clarification on any point, feel free to ask!