Verify the conditions of Rolle's Theorem in the following problems. In each case, find a point in the interval, where the derivative vanishes :
`e^(1-x^(2))" on "[-1,1]`

Answer

To verify the conditions of Rolle's Theorem for the function \( f(x) = e^{1 - x^2} \) on the interval \([-1, 1]\), we will follow these steps:

### Step 1: Check Continuity
We need to check if \( f(x) \) is continuous on the closed interval \([-1, 1]\).

**Solution:**
The function \( f(x) = e^{1 - x^2} \) is an exponential function, and exponential functions are continuous everywhere. Therefore, \( f(x) \) is continuous on \([-1, 1]\).

### Step 2: Check Differentiability
Next, we need to check if \( f(x) \) is differentiable on the open interval \((-1, 1)\).

**Solution:**
Since \( f(x) \) is an exponential function, it is also differentiable everywhere, including the open interval \((-1, 1)\). Thus, \( f(x) \) is differentiable on \((-1, 1)\).

### Step 3: Check Endpoints
Now, we need to check if \( f(-1) = f(1) \).

**Solution:**
Calculate \( f(-1) \) and \( f(1) \):
\[ f(-1) = e^{1 - (-1)^2} = e^{1 - 1} = e^0 = 1 \]
\[ f(1) = e^{1 - 1^2} = e^{1 - 1} = e^0 = 1 \]
Since \( f(-1) = f(1) = 1 \), the third condition is satisfied.

### Step 4: Apply Rolle's Theorem
Since all three conditions of Rolle's Theorem are satisfied, there exists at least one point \( c \) in the interval \((-1, 1)\) such that \( f'(c) = 0 \).

**Solution:**
Now we will find \( f'(x) \):
\[ f'(x) = \frac{d}{dx}(e^{1 - x^2}) = e^{1 - x^2} \cdot \frac{d}{dx}(1 - x^2) = e^{1 - x^2} \cdot (-2x) \]
Setting \( f'(x) = 0 \):
\[ e^{1 - x^2} \cdot (-2x) = 0 \]
Since \( e^{1 - x^2} > 0 \) for all \( x \), we have:
\[ -2x = 0 \implies x = 0 \]

### Conclusion
Thus, the point \( c = 0 \) lies in the interval \([-1, 1]\) where the derivative \( f'(c) = 0 \).

### Summary of Steps
1. **Check Continuity**: \( f(x) \) is continuous on \([-1, 1]\).
2. **Check Differentiability**: \( f(x) \) is differentiable on \((-1, 1)\).
3. **Check Endpoint Values**: \( f(-1) = f(1) = 1 \).
4. **Find \( c \)**: \( c = 0 \) is the point where \( f'(c) = 0 \).