The equation `e^x = m(m +1), m<0` has

Options

A. no real root
B. exactly one real root
C. two real root
D. None of the above

Answer

To solve the equation \( e^x = m(m + 1) \) under the condition \( m < 0 \), we will analyze the expression step by step.

### Step 1: Understand the equation
The equation given is:
\[ e^x = m(m + 1) \]
where \( m < 0 \).

### Step 2: Analyze the right-hand side
The expression \( m(m + 1) \) can be rewritten as:
\[ m^2 + m \]
Since \( m < 0 \), we need to determine the sign of \( m(m + 1) \).

### Step 3: Determine the range of \( m(m + 1) \)
1. **Finding the roots**: The expression \( m(m + 1) = 0 \) has roots at \( m = 0 \) and \( m = -1 \).
2. **Sign analysis**:
- For \( m < -1 \): \( m(m + 1) > 0 \) (both factors are negative, product is positive).
- For \( -1 < m < 0 \): \( m(m + 1) < 0 \) (one factor is negative and the other is positive, product is negative).

### Step 4: Conclusion about \( m(m + 1) \)
Since we are given \( m < 0 \) and specifically looking at \( m < -1 \), we find that:
\[ m(m + 1) > 0 \]
This means \( e^x = m(m + 1) \) is positive.

### Step 5: Logarithmic transformation
Taking the natural logarithm of both sides:
\[ x = \ln(m(m + 1)) \]
Since \( m(m + 1) > 0 \), \( \ln(m(m + 1)) \) is defined.

### Step 6: Determine the number of real roots
The function \( e^x \) is always positive and increases without bound. The right-hand side \( m(m + 1) \) is a constant (for a fixed \( m < -1 \)). Therefore:
- If \( m(m + 1) > 0 \), there will be exactly one solution for \( x \) since the exponential function intersects a horizontal line at one point.

### Final Conclusion
The equation \( e^x = m(m + 1) \) has exactly one real root when \( m < -1 \).

### Answer
The correct option is: **exactly 1 real root**.

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