If complete solution set of `e ^(-x) le 4- x ` is `[alpha, beta]` then `[alpha] + [beta ]` is equal to : (where [.] denotes greatest integer function )

Options

A. 0
B. 2
C. 1
D. 4

Answer

To solve the inequality \( e^{-x} \leq 4 - x \), we will follow these steps:

### Step 1: Rewrite the inequality
We start with the given inequality:
\[ e^{-x} \leq 4 - x \]

### Step 2: Identify the functions
Let \( f(x) = e^{-x} \) and \( g(x) = 4 - x \). We need to find the points where these two functions intersect and the regions where \( f(x) \) is less than or equal to \( g(x) \).

### Step 3: Analyze the functions
- The function \( f(x) = e^{-x} \) is an exponential decay function that approaches 0 as \( x \) increases.
- The function \( g(x) = 4 - x \) is a linear function with a y-intercept of 4 and a slope of -1.

### Step 4: Find intersection points
To find the intersection points, we set \( f(x) = g(x) \):
\[ e^{-x} = 4 - x \]
This equation does not have a simple algebraic solution, so we will estimate the values of \( x \) where these two functions intersect.

### Step 5: Estimate values for \( \alpha \) and \( \beta \)
We will check values of \( x \) to find where \( e^{-x} \) is less than or equal to \( 4 - x \).

1. **Check \( x = -1 \)**:
\[ e^{-(-1)} = e^1 \approx 2.718 \quad \text{and} \quad 4 - (-1) = 5 \]
Since \( 2.718 < 5 \), this point is valid.

2. **Check \( x = -2 \)**:
\[ e^{-(-2)} = e^2 \approx 7.389 \quad \text{and} \quad 4 - (-2) = 6 \]
Since \( 7.389 > 6 \), this point is not valid.

Thus, \( \alpha \) lies between \( -1 \) and \( -2 \).

3. **Check \( x = 1 \)**:
\[ e^{-1} \approx 0.368 \quad \text{and} \quad 4 - 1 = 3 \]
Since \( 0.368 < 3 \), this point is valid.

4. **Check \( x = 2 \)**:
\[ e^{-2} \approx 0.135 \quad \text{and} \quad 4 - 2 = 2 \]
Since \( 0.135 < 2 \), this point is valid.

5. **Check \( x = 3 \)**:
\[ e^{-3} \approx 0.0498 \quad \text{and} \quad 4 - 3 = 1 \]
Since \( 0.0498 < 1 \), this point is valid.

6. **Check \( x = 4 \)**:
\[ e^{-4} \approx 0.0183 \quad \text{and} \quad 4 - 4 = 0 \]
Since \( 0.0183 > 0 \), this point is not valid.

Thus, \( \beta \) lies between \( 3 \) and \( 4 \).

### Step 6: Determine the greatest integer values
- For \( \alpha \), which lies between \( -1 \) and \( -2 \), the greatest integer function \( [\alpha] = -2 \).
- For \( \beta \), which lies between \( 3 \) and \( 4 \), the greatest integer function \( [\beta] = 3 \).

### Step 7: Calculate the final result
Now, we add the greatest integer values:
\[ [\alpha] + [\beta] = -2 + 3 = 1 \]

### Final Answer
The value of \( [\alpha] + [\beta] \) is \( 1 \).

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