Number of solution of `6 abscosx = x " in"[ 0.2pi]` is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the number of solutions to the equation \(6 |\cos x| = x\) in the interval \([0, 2\pi]\), we can follow these steps:

### Step 1: Understand the Functions
We need to analyze the two functions involved in the equation:
1. \(y = 6 |\cos x|\)
2. \(y = x\)

### Step 2: Graph the Functions
- The graph of \(y = |\cos x|\) oscillates between 0 and 1. Therefore, the graph of \(y = 6 |\cos x|\) will oscillate between 0 and 6.
- The graph of \(y = x\) is a straight line that passes through the origin and increases linearly.

### Step 3: Determine the Interval
We are interested in the interval \([0, 2\pi]\). In this interval:
- The maximum value of \(y = 6 |\cos x|\) is 6 (occurs at \(x = 0, \pi, 2\pi\)).
- The value of \(y = x\) at \(x = 2\pi\) is approximately 6.28.

### Step 4: Identify Intersections
To find the number of solutions, we need to determine how many times the graph of \(y = 6 |\cos x|\) intersects with the line \(y = x\) in the interval \([0, 2\pi]\).

1. At \(x = 0\), \(y = 6 |\cos(0)| = 6\) and \(y = 0\), so they intersect.
2. As \(x\) increases to \(\frac{\pi}{2}\), \(y = 6 |\cos x|\) decreases to 0, while \(y = x\) increases. They intersect once between \(0\) and \(\frac{\pi}{2}\).
3. From \(\frac{\pi}{2}\) to \(\frac{3\pi}{2}\), \(y = 6 |\cos x|\) increases again, reaching a maximum of 6 at \(x = \pi\). They intersect again between \(\frac{\pi}{2}\) and \(\pi\).
4. Finally, from \(\frac{3\pi}{2}\) to \(2\pi\), \(y = 6 |\cos x|\) decreases back to 0, while \(y = x\) continues to increase. They intersect once more between \(\pi\) and \(2\pi\).

### Step 5: Count the Intersections
From the analysis:
- There are three points of intersection in total:
1. Between \(0\) and \(\frac{\pi}{2}\)
2. Between \(\frac{\pi}{2}\) and \(\pi\)
3. Between \(\pi\) and \(2\pi\)

### Conclusion
Thus, the number of solutions to the equation \(6 |\cos x| = x\) in the interval \([0, 2\pi]\) is **3**.

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