The number of distinct real roots of the equation `x=((5pi)/(2))^(cos x)`

Options

A. (a) 0
B. (b) 1
C. (c) 2
D. (d) 3

Explanation

To find the number of distinct real roots of the equation \( x = \left( \frac{5\pi}{2} \right)^{\cos x} \), we can follow these steps:

### Step 1: Take the logarithm of both sides
We start by taking the logarithm of both sides of the equation:

\[ \log x = \log\left(\left(\frac{5\pi}{2}\right)^{\cos x}\right) \]

### Step 2: Apply the power rule of logarithms
Using the power rule of logarithms, we can rewrite the right-hand side:

\[ \log x = \cos x \cdot \log\left(\frac{5\pi}{2}\right) \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation to isolate \(\cos x\):

\[ \cos x = \frac{\log x}{\log\left(\frac{5\pi}{2}\right)} \]

### Step 4: Define the functions
Let \( y_1 = \cos x \) and \( y_2 = \frac{\log x}{\log\left(\frac{5\pi}{2}\right)} \). We need to find the number of intersections between these two functions.

### Step 5: Analyze the function \( y_1 = \cos x \)
The function \( y_1 = \cos x \) oscillates between -1 and 1 with a period of \( 2\pi \).

### Step 6: Analyze the function \( y_2 = \frac{\log x}{\log\left(\frac{5\pi}{2}\right)} \)
The function \( y_2 \) is defined for \( x > 0 \) and is an increasing function since the logarithm is increasing. At \( x = 1 \):

\[ y_2(1) = \frac{\log 1}{\log\left(\frac{5\pi}{2}\right)} = 0 \]

At \( x = \frac{5\pi}{2} \):

\[ y_2\left(\frac{5\pi}{2}\right) = \frac{\log\left(\frac{5\pi}{2}\right)}{\log\left(\frac{5\pi}{2}\right)} = 1 \]

### Step 7: Determine the behavior of \( y_2 \)
As \( x \to 0^+ \), \( y_2 \to -\infty \), and as \( x \to \infty \), \( y_2 \to \infty \). Therefore, \( y_2 \) crosses the x-axis at \( x = 1 \) and reaches \( y = 1 \) at \( x = \frac{5\pi}{2} \).

### Step 8: Find intersections
To find the number of intersections between \( y_1 \) and \( y_2 \):

1. **From \( x = 0 \) to \( x = 1 \)**: \( y_1 \) starts at 1 and decreases to 0, while \( y_2 \) starts from \(-\infty\) and increases to 0. There is one intersection.
2. **From \( x = 1 \) to \( x = \frac{5\pi}{2} \)**: \( y_1 \) oscillates between 1 and -1, while \( y_2 \) increases from 0 to 1. There are two intersections (one when \( y_1 \) is decreasing and one when it is increasing).
3. **From \( x = \frac{5\pi}{2} \) onwards**: \( y_1 \) continues to oscillate, and \( y_2 \) continues to increase. There is one more intersection as \( y_2 \) will eventually exceed the maximum value of \( y_1 \).

### Conclusion
In total, we find 3 distinct intersections. Therefore, the number of distinct real roots of the equation \( x = \left( \frac{5\pi}{2} \right)^{\cos x} \) is:

\[ \boxed{3} \]