If the Rolle's theorem is applicable to the function f defined by
`f(x)={{:(ax^(2)+b",", xle1),(1",", x=1),((c)/(|x|)",",xgt1):}`
in the interval `[-3, 3]`, then which of the following alternative(s) is/are correct?

Options

A. `a+b+c=2`
B. `|a|+|b|+|c|=3`
C. `2a+4b+3c=8`
D. `4a^(2)+4b^(2)+5c^(2)=15`

Answer

To determine if Rolle's theorem is applicable to the function \( f \) defined as follows:

\[ f(x) = \begin{cases} ax^2 + b & \text{for } x \leq 1 \\ 1 & \text{for } x = 1 \\ \frac{c}{|x|} & \text{for } x > 1 \end{cases} \]

in the interval \([-3, 3]\), we need to check the conditions for Rolle's theorem, which states that if a function is continuous on a closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), and \( f(a) = f(b) \), then there exists at least one \( c \) in \((a, b)\) such that \( f'(c) = 0 \).

### Step 1: Check Continuity at \( x = 1 \)

To apply Rolle's theorem, we first need to ensure that \( f \) is continuous at \( x = 1 \).

1. **Left-hand limit as \( x \to 1^- \)**:
\[ \lim_{x \to 1^-} f(x) = a(1)^2 + b = a + b \]

2. **Value at \( x = 1 \)**:
\[ f(1) = 1 \]

3. **Right-hand limit as \( x \to 1^+ \)**:
\[ \lim_{x \to 1^+} f(x) = \frac{c}{|1|} = c \]

For continuity at \( x = 1 \):
\[ a + b = 1 \quad \text{(1)} \]
\[ c = 1 \quad \text{(2)} \]

### Step 2: Check Differentiability at \( x = 1 \)

Next, we need to check if \( f \) is differentiable at \( x = 1 \).

1. **Left-hand derivative**:
\[ f'(x) = 2ax \quad \Rightarrow \quad f'(1) = 2a \]

2. **Right-hand derivative**:
\[ f'(x) = -\frac{c}{x^2} \quad \Rightarrow \quad f'(1) = -c \]

Setting the left-hand derivative equal to the right-hand derivative:
\[ 2a = -c \quad \text{(3)} \]

### Step 3: Solve the Equations

From equations (1) and (2):
1. From (2): \( c = 1 \)
2. Substitute \( c = 1 \) into (3):
\[ 2a = -1 \quad \Rightarrow \quad a = -\frac{1}{2} \]

3. Substitute \( a = -\frac{1}{2} \) into (1):
\[ -\frac{1}{2} + b = 1 \quad \Rightarrow \quad b = 1 + \frac{1}{2} = \frac{3}{2} \]

### Step 4: Calculate \( a + b + c \)

Now we can calculate:
\[ a + b + c = -\frac{1}{2} + \frac{3}{2} + 1 = 2 \]

### Step 5: Calculate \( |\text{a}| + |\text{b}| + |\text{c}| \)

Calculating the sum of the absolute values:
\[ |a| + |b| + |c| = \left| -\frac{1}{2} \right| + \left| \frac{3}{2} \right| + |1| = \frac{1}{2} + \frac{3}{2} + 1 = 3 \]

### Step 6: Calculate \( 2a + 4b + 3c \)

Calculating:
\[ 2a + 4b + 3c = 2(-\frac{1}{2}) + 4(\frac{3}{2}) + 3(1) = -1 + 6 + 3 = 8 \]

### Step 7: Calculate \( 4a^2 + 4b^2 + 5c^2 \)

Calculating:
\[ 4a^2 + 4b^2 + 5c^2 = 4\left(-\frac{1}{2}\right)^2 + 4\left(\frac{3}{2}\right)^2 + 5(1)^2 = 4 \cdot \frac{1}{4} + 4 \cdot \frac{9}{4} + 5 = 1 + 9 + 5 = 15 \]

### Conclusion

Thus, the values we have calculated are:
- \( a + b + c = 2 \)
- \( |a| + |b| + |c| = 3 \)
- \( 2a + 4b + 3c = 8 \)
- \( 4a^2 + 4b^2 + 5c^2 = 15 \)

All the options provided in the question are correct.