Let `f(x)="min"{sqrt(4-x^(2)),sqrt(1+x^(2))}AA,x in [-2, 2]` then the number of points where `f(x)` is non - differentiable is

Options

A. 1
B. 0
C. 4
D. 2

Answer

To determine the number of points where the function \( f(x) = \min\{\sqrt{4 - x^2}, \sqrt{1 + x^2}\} \) is non-differentiable for \( x \in [-2, 2] \), we will follow these steps:

### Step 1: Identify the functions
The function \( f(x) \) is defined as the minimum of two functions:
1. \( g_1(x) = \sqrt{4 - x^2} \)
2. \( g_2(x) = \sqrt{1 + x^2} \)

### Step 2: Determine the domain of each function
- The function \( g_1(x) = \sqrt{4 - x^2} \) is defined for \( x \in [-2, 2] \) since the expression under the square root must be non-negative.
- The function \( g_2(x) = \sqrt{1 + x^2} \) is defined for all \( x \) since \( 1 + x^2 \) is always positive.

### Step 3: Find the points of intersection
To find where \( f(x) \) may change from one function to the other, we set \( g_1(x) = g_2(x) \):
\[ \sqrt{4 - x^2} = \sqrt{1 + x^2} \]
Squaring both sides gives:
\[ 4 - x^2 = 1 + x^2 \]
Rearranging this equation:
\[ 4 - 1 = 2x^2 \implies 3 = 2x^2 \implies x^2 = \frac{3}{2} \implies x = \pm \sqrt{\frac{3}{2}} \approx \pm 1.2247 \]

### Step 4: Evaluate the function at the endpoints and points of intersection
We need to evaluate \( f(x) \) at the points \( x = -2, -\sqrt{\frac{3}{2}}, \sqrt{\frac{3}{2}}, 2 \):
- At \( x = -2 \):
\[ f(-2) = \min\{\sqrt{4 - (-2)^2}, \sqrt{1 + (-2)^2}\} = \min\{0, \sqrt{5}\} = 0 \]
- At \( x = 2 \):
\[ f(2) = \min\{\sqrt{4 - 2^2}, \sqrt{1 + 2^2}\} = \min\{0, \sqrt{5}\} = 0 \]
- At \( x = -\sqrt{\frac{3}{2}} \):
\[ f(-\sqrt{\frac{3}{2}}) = \sqrt{4 - \frac{3}{2}} = \sqrt{\frac{5}{2}} \quad \text{and} \quad f(-\sqrt{\frac{3}{2}}) = \sqrt{1 + \frac{3}{2}} = \sqrt{\frac{5}{2}} \]
- At \( x = \sqrt{\frac{3}{2}} \):
\[ f(\sqrt{\frac{3}{2}}) = \sqrt{4 - \frac{3}{2}} = \sqrt{\frac{5}{2}} \quad \text{and} \quad f(\sqrt{\frac{3}{2}}) = \sqrt{1 + \frac{3}{2}} = \sqrt{\frac{5}{2}} \]

### Step 5: Check differentiability at the points of intersection
To check for non-differentiability, we need to analyze the left-hand and right-hand limits at the points \( x = -\sqrt{\frac{3}{2}} \) and \( x = \sqrt{\frac{3}{2}} \):
- At \( x = -\sqrt{\frac{3}{2}} \):
- The left-hand limit will approach \( g_1(x) \) and the right-hand limit will approach \( g_2(x) \).
- At \( x = \sqrt{\frac{3}{2}} \):
- The left-hand limit will approach \( g_2(x) \) and the right-hand limit will approach \( g_1(x) \).

### Conclusion
The function \( f(x) \) is non-differentiable at:
1. \( x = -\sqrt{\frac{3}{2}} \)
2. \( x = \sqrt{\frac{3}{2}} \)
3. \( x = -2 \)
4. \( x = 2 \)

Thus, there are **4 points** where \( f(x) \) is non-differentiable.