Given the function $f(x) = \cos\left(e^2\right) \cdot [x] + \cos\left(-e^2\right) \cdot [x]$, where $[x]$ represents the greatest integer function (also known as the floor function), we need to analyze the value of $f(x)$ for different values of $x$.

Step 1: Simplify the Function

We start by recognizing that $\cos\left(e^2\right) = \cos\left(-e^2\right)$ because the cosine function is an even function, meaning $\cos(-\theta) = \cos(\theta)$. So the function simplifies to:

[ f(x) = \cos\left(e^2\right) \cdot [x] + \cos\left(e^2\right) \cdot [x] ] $f(x) = 2 \cdot \cos\left(e^2\right) \cdot [x]$

Step 2: Evaluate $f(x)$ for Specific Values

Let's evaluate the function for the given specific values of $x$.

1. For $x = \pi$: $f(\pi) = 2 \cdot \cos\left(e^2\right) \cdot [\pi]$ Since $\pi$ is approximately 3.14159, $[\pi] = 3$: $f(\pi) = 2 \cdot \cos\left(e^2\right) \cdot 3$ This is not necessarily equal to 1 unless $\cos\left(e^2\right)$ is a specific value, but it's not generally 1.

2. For $x = 2\pi$: $f(2\pi) = 2 \cdot \cos\left(e^2\right) \cdot [2\pi]$ Since $2\pi$ is approximately 6.28318, $[2\pi] = 6$: $f(2\pi) = 2 \cdot \cos\left(e^2\right) \cdot 6$ Again, this is not necessarily equal to 1.

3. For $x = \frac{\pi}{2}$: $f\left(\frac{\pi}{2}\right) = 2 \cdot \cos\left(e^2\right) \cdot \left[\frac{\pi}{2}\right]$ Since $\frac{\pi}{2}$ is approximately 1.5708, $\left[\frac{\pi}{2}\right] = 1$: $f\left(\frac{\pi}{2}\right) = 2 \cdot \cos\left(e^2\right) \cdot 1 = 2 \cdot \cos\left(e^2\right)$ This is not necessarily equal to 1.

4. For $x = \frac{\pi}{4}$: $f\left(\frac{\pi}{4}\right) = 2 \cdot \cos\left(e^2\right) \cdot \left[\frac{\pi}{4}\right]$ Since $\frac{\pi}{4}$ is approximately 0.7854, $\left[\frac{\pi}{4}\right] = 0$: $f\left(\frac{\pi}{4}\right) = 2 \cdot \cos\left(e^2\right) \cdot 0 = 0$ This is certainly not equal to 1.

Conclusion

From this analysis:

• $f(\pi) = 6 \cdot \cos(e^2)$
• $f(2\pi) = 12 \cdot \cos(e^2)$
• $f(\frac{\pi}{2}) = 2 \cdot \cos(e^2)$
• $f(\frac{\pi}{4}) = 0$

None of these values are guaranteed to be 1 under general circumstances, so none of the options given are correct.

Would you like any further details or explanations?

Related Questions

1. What is the greatest integer function, and how does it work?
2. How does the even nature of the cosine function affect the simplification of functions?
3. What is the value of $\cos(e^2)$ approximately?
4. Can $f(x)$ be 1 for some specific value of $x$?
5. What other properties of the floor function are useful in simplifying expressions?
6. How do you calculate the floor function for various values?
7. How does the choice of $x$ affect the value of $f(x)$?
8. Can you find a value of $x$ for which $f(x) = 1$?

Tip: Always check if the function in a trigonometric expression simplifies due to the properties of the trigonometric functions.