The total number of solutions of `log_e |sin x| = -x^2 +2x in [0,pi]` is equal to

Options

A. 1
B. 2
C. 4
D. none of these

Explanation

To solve the equation \( \log_e |\sin x| = -x^2 + 2x \) in the interval \( [0, \pi] \), we will analyze both sides of the equation step by step.

### Step 1: Define the Functions
Let:
- \( f(x) = \log_e |\sin x| \)
- \( g(x) = -x^2 + 2x \)

We need to find the number of solutions to the equation \( f(x) = g(x) \) in the interval \( [0, \pi] \).

### Step 2: Analyze \( f(x) = \log_e |\sin x| \)
The function \( f(x) \) is defined for \( x \in (0, \pi) \) because \( \sin x \) is positive in this interval. At the endpoints:
- As \( x \to 0^+ \), \( \sin x \to 0 \) and thus \( f(x) \to -\infty \).
- At \( x = \pi \), \( \sin x = 0 \) again, so \( f(x) \to -\infty \).

The maximum value of \( f(x) \) occurs at \( x = \frac{\pi}{2} \):
- \( f\left(\frac{\pi}{2}\right) = \log_e(1) = 0 \).

### Step 3: Analyze \( g(x) = -x^2 + 2x \)
The function \( g(x) \) is a downward-opening parabola. We can rewrite it as:
\[ g(x) = - (x^2 - 2x) = - (x(x - 2)) \]
The roots of \( g(x) \) are \( x = 0 \) and \( x = 2 \). The vertex of the parabola occurs at:
\[ x = -\frac{b}{2a} = \frac{2}{2} = 1 \]
Calculating \( g(1) \):
\[ g(1) = -1^2 + 2 \cdot 1 = 1 \]

### Step 4: Determine the Range of \( g(x) \)
- At \( x = 0 \), \( g(0) = 0 \).
- At \( x = 2 \), \( g(2) = 0 \).
- The maximum value of \( g(x) \) is \( g(1) = 1 \).

Thus, \( g(x) \) ranges from \( 0 \) to \( 1 \) in the interval \( [0, 2] \).

### Step 5: Compare the Functions
Now we compare the ranges of \( f(x) \) and \( g(x) \):
- \( f(x) \) ranges from \( -\infty \) (as \( x \to 0^+ \) and \( x = \pi \)) to \( 0 \) (at \( x = \frac{\pi}{2} \)).
- \( g(x) \) ranges from \( 0 \) (at \( x = 0 \) and \( x = 2 \)) to \( 1 \) (at \( x = 1 \)).

### Step 6: Find the Number of Solutions
Since \( f(x) \) reaches a maximum of \( 0 \) and \( g(x) \) starts at \( 0 \) and goes up to \( 1 \), the two functions intersect at:
- \( f(x) = g(x) = 0 \) at \( x = 0 \) and \( x = \pi \), but \( f(x) \) is not defined at \( x = 0 \) and \( x = \pi \) (as \( \log_e(0) \) is undefined).

Thus, there are **no solutions** in the interval \( [0, \pi] \).

### Final Answer
The total number of solutions of \( \log_e |\sin x| = -x^2 + 2x \) in the interval \( [0, \pi] \) is **0**.