Statement 1: `f(x)=x^(3)` is a one-one function.
Statement 2: Any monotonic function is a one-one function.

Options

A. (a) Statement 1 and Statement 2 are true and Statement 2 is the correct explanation for Statement 1.
B. (b) Statement 1 and Statement 2 are true but Statement 2 is not the correct explanation for Statement 1.
C. (c) Statement 1 is true but Statement 2 is false
D. (d) Statement 2 is true but Statement 1 is false

Answer

To solve the problem, we need to analyze both statements regarding the function \( f(x) = x^3 \).

### Step 1: Verify Statement 1
We need to determine if \( f(x) = x^3 \) is a one-one function. A function is one-one (injective) if for any two values \( x_1 \) and \( x_2 \), if \( f(x_1) = f(x_2) \), then it must imply that \( x_1 = x_2 \).

1. Assume \( f(x_1) = f(x_2) \).
\[ x_1^3 = x_2^3 \]

2. Taking the cube root of both sides, we get:
\[ x_1 = x_2 \]

Since \( x_1 = x_2 \) whenever \( f(x_1) = f(x_2) \), we conclude that \( f(x) = x^3 \) is indeed a one-one function.

### Step 2: Verify Statement 2
Statement 2 states that any monotonic function is a one-one function. A function is monotonic if it is either entirely non-increasing or non-decreasing.

1. To check if \( f(x) = x^3 \) is monotonic, we can find its derivative:
\[ f'(x) = 3x^2 \]

2. The derivative \( f'(x) = 3x^2 \) is always greater than or equal to 0 for all \( x \) (since squares are non-negative). Specifically, \( f'(x) > 0 \) for all \( x \neq 0 \).

3. Since the derivative is non-negative and only equals zero at \( x = 0 \), the function is monotonically increasing everywhere.

### Conclusion
Since \( f(x) = x^3 \) is a one-one function and it is also a monotonic function (specifically, monotonically increasing), both statements are true. Additionally, Statement 2 correctly explains Statement 1.

### Final Answer
Both Statement 1 and Statement 2 are true, and Statement 2 is a correct explanation for Statement 1.