Let `f(x) = {:(1/(x^2) , : |x| ge 1),(alphax^2 + beta , : |x| < 1):}` . If f(x) is continuous and differentiable at any point, then

Options

A. `alpha = 2, beta = 1`
B. `alpha = -1, beta = 2`
C. `alpha = 1, beta = 0`
D. `alpha = -2, beta = 3`

Answer

To solve the problem, we need to find the values of α and β such that the function \( f(x) \) is continuous and differentiable at all points, particularly at the points where the definition of the function changes, which are \( x = 1 \) and \( x = -1 \).

The function is defined as:
\[ f(x) = \begin{cases} \frac{1}{x^2} & \text{for } |x| \geq 1 \\ \alpha x^2 + \beta & \text{for } |x| < 1 \end{cases} \]

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

For \( f(x) \) to be continuous at \( x = 1 \), we need:
\[ f(1^-) = f(1^+) \]

Calculating \( f(1^-) \):
\[ f(1^-) = \alpha(1^2) + \beta = \alpha + \beta \]

Calculating \( f(1^+) \):
\[ f(1^+) = \frac{1}{1^2} = 1 \]

Setting these equal for continuity:
\[ \alpha + \beta = 1 \quad \text{(Equation 1)} \]

### Step 2: Ensure Continuity at \( x = -1 \)

For \( f(x) \) to be continuous at \( x = -1 \), we need:
\[ f(-1^-) = f(-1^+) \]

Calculating \( f(-1^-) \):
\[ f(-1^-) = \alpha(-1^2) + \beta = \alpha + \beta \]

Calculating \( f(-1^+) \):
\[ f(-1^+) = \frac{1}{(-1)^2} = 1 \]

Setting these equal for continuity:
\[ \alpha + \beta = 1 \quad \text{(This is the same as Equation 1)} \]

### Step 3: Ensure Differentiability at \( x = 1 \)

For \( f(x) \) to be differentiable at \( x = 1 \), we need:
\[ f'(1^-) = f'(1^+) \]

Calculating \( f'(1^-) \):
\[ f'(x) = \frac{d}{dx}(\alpha x^2 + \beta) = 2\alpha x \]
Thus,
\[ f'(1^-) = 2\alpha(1) = 2\alpha \]

Calculating \( f'(1^+) \):
\[ f'(x) = \frac{d}{dx}\left(\frac{1}{x^2}\right) = -\frac{2}{x^3} \]
Thus,
\[ f'(1^+) = -\frac{2}{1^3} = -2 \]

Setting these equal for differentiability:
\[ 2\alpha = -2 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations

From Equation 2:
\[ 2\alpha = -2 \implies \alpha = -1 \]

Substituting \( \alpha = -1 \) into Equation 1:
\[ -1 + \beta = 1 \implies \beta = 2 \]

### Conclusion

The values of \( \alpha \) and \( \beta \) are:
\[ \alpha = -1, \quad \beta = 2 \]