Using Rolle's theorem , find a point on the curve `y = x^(2) , x in [-1,1]` at which the tangent is parallel to X-axis.

Answer

To find a point on the curve \( y = x^2 \) where the tangent is parallel to the x-axis using Rolle's theorem, we can follow these steps:

### Step 1: Verify the conditions of Rolle's Theorem
Rolle's theorem states that if a function \( f(x) \) is continuous on the closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and \( f(a) = f(b) \), then there exists at least one point \( c \) in \((a, b)\) such that \( f'(c) = 0 \).

1. **Function Definition**: Let \( f(x) = x^2 \).
2. **Interval**: We are considering the interval \([-1, 1]\).

### Step 2: Check continuity
The function \( f(x) = x^2 \) is a polynomial, and polynomials are continuous everywhere. Therefore, \( f(x) \) is continuous on the interval \([-1, 1]\).

### Step 3: Check differentiability
The function \( f(x) = x^2 \) is also differentiable everywhere, including the open interval \((-1, 1)\).

### Step 4: Check the endpoints
Now we need to check the values of the function at the endpoints of the interval:
- \( f(-1) = (-1)^2 = 1 \)
- \( f(1) = (1)^2 = 1 \)

Since \( f(-1) = f(1) \), the third condition of Rolle's theorem is satisfied.

### Step 5: Apply Rolle's Theorem
Since all conditions of Rolle's theorem are satisfied, there exists at least one point \( c \) in the interval \((-1, 1)\) such that \( f'(c) = 0 \).

### Step 6: Find the derivative
To find \( c \), we first calculate the derivative of \( f(x) \):
\[ f'(x) = \frac{d}{dx}(x^2) = 2x \]

### Step 7: Set the derivative to zero
Now, we set the derivative equal to zero to find the point where the tangent is parallel to the x-axis:
\[ 2c = 0 \]
\[ c = 0 \]

### Step 8: Conclusion
Thus, the point on the curve \( y = x^2 \) where the tangent is parallel to the x-axis is at \( x = 0 \). The corresponding \( y \) value is:
\[ y = f(0) = 0^2 = 0 \]
So the point is \( (0, 0) \).

### Summary
The point on the curve \( y = x^2 \) at which the tangent is parallel to the x-axis is \( (0, 0) \).

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