Rolle's theorem in applicable in the interval [-1,1] for the function

Options

A. f(x)=x
B. `f(x)=x^(2)`
C. `f(x)=x^(3)+2`
D. `f(x)=|x|`

Answer

To determine if Rolle's Theorem is applicable to the given function in the interval \([-1, 1]\), we need to follow the steps outlined by the theorem itself. Here’s a step-by-step solution:

### Step 1: State the conditions of Rolle's Theorem
Rolle's Theorem states that if a function \( f \) satisfies the following conditions on the interval \([a, b]\):
1. \( f \) is continuous on the closed interval \([a, b]\),
2. \( f \) is differentiable on the open interval \((a, b)\),
3. \( f(a) = f(b) \),

Then there exists at least one point \( c \) in the open interval \((a, b)\) such that \( f'(c) = 0 \).

### Step 2: Identify the function and the interval
We are given the interval \([-1, 1]\). We need to check the provided options to find a function that meets the conditions of Rolle's Theorem. Let's consider option (b) where \( f(x) = x^2 \).

### Step 3: Check continuity
To check if \( f(x) = x^2 \) is continuous on \([-1, 1]\):
- The function \( f(x) = x^2 \) is a polynomial, and all polynomials are continuous everywhere. Therefore, \( f(x) \) is continuous on \([-1, 1]\).

### Step 4: Check differentiability
Next, we check if \( f(x) = x^2 \) is differentiable on \((-1, 1)\):
- The derivative \( f'(x) = 2x \) is also a polynomial, which is differentiable everywhere. Thus, \( f(x) \) is differentiable on \((-1, 1)\).

### Step 5: Check the endpoints
Now we need to check if \( f(-1) = f(1) \):
- Calculate \( f(-1) = (-1)^2 = 1 \) and \( f(1) = (1)^2 = 1 \).
- Since \( f(-1) = f(1) \), the condition \( f(a) = f(b) \) is satisfied.

### Step 6: Apply Rolle's Theorem
Since all conditions of Rolle's Theorem are satisfied, we can conclude that there exists at least one point \( c \) in the interval \((-1, 1)\) such that:
\[ f'(c) = 0 \]

### Step 7: Find the point \( c \)
To find \( c \):
- Set the derivative equal to zero:
\[ 2c = 0 \]
- Solving this gives \( c = 0 \), which lies in the interval \((-1, 1)\).

### Conclusion
Thus, the function \( f(x) = x^2 \) satisfies all the conditions of Rolle's Theorem in the interval \([-1, 1]\), and the correct option is (b).

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