A function is defined as follows :
`f(x) = {{:(x^(3), x^(2) lt 1),(x, x^(2) ge 1):}`
The function is:

Options

A. continuous at x = 1
B. differentiable at x = 1
C. continuous but not differentiable at x = 1
D. none of these

Answer

To analyze the function \( f(x) \) defined as follows:

\[ f(x) = \begin{cases} x^3 & \text{if } x^2 < 1 \\ x & \text{if } x^2 \geq 1 \end{cases} \]

we need to check its continuity and differentiability at the point \( x = 1 \).

### Step 1: Determine the left-hand limit as \( x \) approaches 1.

For \( x < 1 \), we use the first case of the function, which is \( f(x) = x^3 \).

\[ \lim_{h \to 0} f(1 - h) = \lim_{h \to 0} (1 - h)^3 \]

Calculating this limit:

\[ (1 - h)^3 = 1 - 3h + 3h^2 - h^3 \]

Taking the limit as \( h \to 0 \):

\[ \lim_{h \to 0} (1 - 3h + 3h^2 - h^3) = 1 \]

### Step 2: Determine the right-hand limit as \( x \) approaches 1.

For \( x \geq 1 \), we use the second case of the function, which is \( f(x) = x \).

\[ \lim_{h \to 0} f(1 + h) = \lim_{h \to 0} (1 + h) = 1 \]

### Step 3: Evaluate the function at \( x = 1 \).

Since \( x = 1 \) falls under the condition \( x^2 \geq 1 \):

\[ f(1) = 1 \]

### Step 4: Check continuity at \( x = 1 \).

A function is continuous at a point if:

\[ \lim_{x \to c} f(x) = f(c) \]

Here, we have:

\[ \lim_{x \to 1^-} f(x) = 1, \quad \lim_{x \to 1^+} f(x) = 1, \quad f(1) = 1 \]

Since all these values are equal, \( f(x) \) is continuous at \( x = 1 \).

### Step 5: Check differentiability at \( x = 1 \).

To check differentiability, we need to find the left-hand derivative and the right-hand derivative at \( x = 1 \).

**Left-hand derivative:**

\[ f'(1) = \lim_{h \to 0} \frac{f(1 - h) - f(1)}{-h} = \lim_{h \to 0} \frac{(1 - h)^3 - 1}{-h} \]

Calculating this:

\[ (1 - h)^3 - 1 = -3h + 3h^2 - h^3 \]

So,

\[ \lim_{h \to 0} \frac{-3h + 3h^2 - h^3}{-h} = \lim_{h \to 0} (3 - 3h + h^2) = 3 \]

**Right-hand derivative:**

\[ f'(1) = \lim_{h \to 0} \frac{f(1 + h) - f(1)}{h} = \lim_{h \to 0} \frac{(1 + h) - 1}{h} = \lim_{h \to 0} 1 = 1 \]

### Conclusion:

Since the left-hand derivative (3) does not equal the right-hand derivative (1), the function \( f(x) \) is not differentiable at \( x = 1 \).

### Final Answer:

The function \( f(x) \) is continuous at \( x = 1 \) but not differentiable at \( x = 1 \).

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