Examine the derivability of the following functions at the specified points :
`|x|^(2)" at "x = 0`

Answer

To examine the derivability of the function \( f(x) = |x|^2 \) at the point \( x = 0 \), we will follow these steps:

### Step 1: Define the function
The function is given as:
\[ f(x) = |x|^2 \]
This can be rewritten as:
\[ f(x) = \begin{cases} x^2 & \text{if } x \geq 0 \\ (-x)^2 & \text{if } x < 0 \end{cases} \]
Both cases simplify to \( f(x) = x^2 \).

### Step 2: Check continuity at \( x = 0 \)
To check if \( f(x) \) is continuous at \( x = 0 \), we need to evaluate:
- \( f(0) \)
- \( \lim_{x \to 0} f(x) \)

Calculating \( f(0) \):
\[ f(0) = |0|^2 = 0 \]

Calculating the limit as \( x \) approaches 0:
\[ \lim_{x \to 0} f(x) = \lim_{x \to 0} |x|^2 = |0|^2 = 0 \]

Since \( f(0) = \lim_{x \to 0} f(x) = 0 \), the function is continuous at \( x = 0 \).

### Step 3: Check differentiability at \( x = 0 \)
To check if \( f(x) \) is differentiable at \( x = 0 \), we need to compute the derivative using the definition:
\[ f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h} \]

Substituting \( f(0) = 0 \):
\[ f'(0) = \lim_{h \to 0} \frac{f(h)}{h} = \lim_{h \to 0} \frac{|h|^2}{h} \]

This can be simplified:
\[ f'(0) = \lim_{h \to 0} |h| = 0 \]

### Conclusion
Since \( f'(0) = 0 \), the function \( f(x) = |x|^2 \) is differentiable at \( x = 0 \).

### Final Answer
The function \( f(x) = |x|^2 \) is derivable at \( x = 0 \).

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