The equation `3 cos^(-1) x - pi x - (pi)/(2) =0`has

Options

A. one negative solution
B. one positive solution
C. no solution
D. more than one solution

Explanation

To solve the equation \( 3 \cos^{-1} x - \pi x - \frac{\pi}{2} = 0 \), we will follow these steps:

### Step 1: Rearranging the Equation
We start by rearranging the equation to isolate \( \cos^{-1} x \):
\[ 3 \cos^{-1} x = \pi x + \frac{\pi}{2} \]
Now, divide both sides by 3:
\[ \cos^{-1} x = \frac{\pi x}{3} + \frac{\pi}{6} \]

### Step 2: Understanding the Range of \( \cos^{-1} x \)
The function \( \cos^{-1} x \) has a range of \( [0, \pi] \) for \( x \) in the domain \( [-1, 1] \). Therefore, we need to check if the right-hand side, \( \frac{\pi x}{3} + \frac{\pi}{6} \), lies within this range.

### Step 3: Finding the Range of the Right-Hand Side
To find the range of \( \frac{\pi x}{3} + \frac{\pi}{6} \):
1. When \( x = -1 \):
\[ \frac{\pi(-1)}{3} + \frac{\pi}{6} = -\frac{\pi}{3} + \frac{\pi}{6} = -\frac{2\pi}{6} + \frac{\pi}{6} = -\frac{\pi}{6} \]
2. When \( x = 1 \):
\[ \frac{\pi(1)}{3} + \frac{\pi}{6} = \frac{\pi}{3} + \frac{\pi}{6} = \frac{2\pi}{6} + \frac{\pi}{6} = \frac{3\pi}{6} = \frac{\pi}{2} \]

Thus, the range of \( \frac{\pi x}{3} + \frac{\pi}{6} \) as \( x \) varies from -1 to 1 is \( \left[-\frac{\pi}{6}, \frac{\pi}{2}\right] \).

### Step 4: Finding Intersection Points
Now, we need to find the intersection points of \( y = \cos^{-1} x \) and \( y = \frac{\pi x}{3} + \frac{\pi}{6} \):
- The left side \( y = \cos^{-1} x \) ranges from \( \pi \) (when \( x = -1 \)) to \( 0 \) (when \( x = 1 \)).
- The right side \( y = \frac{\pi x}{3} + \frac{\pi}{6} \) ranges from \( -\frac{\pi}{6} \) to \( \frac{\pi}{2} \).

### Step 5: Analyzing the Graphs
Since \( \cos^{-1} x \) is a decreasing function and \( \frac{\pi x}{3} + \frac{\pi}{6} \) is an increasing linear function, they will intersect at most once in the interval where both functions are defined.

### Conclusion
Given that the two graphs intersect at one point, we conclude that the equation \( 3 \cos^{-1} x - \pi x - \frac{\pi}{2} = 0 \) has exactly **one solution**.

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