Consider the following statements
1 f(x)=In x is an increasing funciton on `(0,oo)`
`2 f(x) =e^(x)-x(In x)` is an increasing function on `(1,oo)`
Which of the above statement is / are correct ?

Options

A. 1 only
B. 2 only
C. both 1 and 2
D. neither 1 nor 2

Explanation

To determine the correctness of the given statements, we will analyze each statement one by one.

### Statement 1: \( f(x) = \ln x \) is an increasing function on \( (0, \infty) \).

**Step 1: Find the derivative of \( f(x) \).**
\[ f'(x) = \frac{d}{dx}(\ln x) = \frac{1}{x} \]

**Step 2: Analyze the derivative.**
- The derivative \( f'(x) = \frac{1}{x} \) is positive for all \( x > 0 \).
- Since \( f'(x) > 0 \) for \( x \in (0, \infty) \), it implies that \( f(x) = \ln x \) is an increasing function on this interval.

**Conclusion for Statement 1:**
- The statement is **true**.

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### Statement 2: \( f(x) = e^x - x \ln x \) is an increasing function on \( (1, \infty) \).

**Step 1: Find the derivative of \( f(x) \).**
\[ f'(x) = \frac{d}{dx}(e^x - x \ln x) \]
Using the product rule on \( x \ln x \):
\[ f'(x) = e^x - \left( \ln x + 1 \right) \]

**Step 2: Analyze the derivative.**
- We need to check if \( f'(x) > 0 \) for \( x \in (1, \infty) \).
- Simplifying, we have:
\[ f'(x) = e^x - \ln x - 1 \]

**Step 3: Evaluate at a specific point, say \( x = 1 \).**
\[ f'(1) = e^1 - \ln(1) - 1 = e - 0 - 1 = e - 1 \]
Since \( e \approx 2.718 \), we have \( f'(1) > 0 \).

**Step 4: Analyze the behavior as \( x \) increases.**
- As \( x \) increases, \( e^x \) grows exponentially while \( \ln x + 1 \) grows much slower.
- Therefore, \( f'(x) = e^x - \ln x - 1 \) will remain positive for \( x > 1 \).

**Conclusion for Statement 2:**
- The statement is **true**.

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### Final Conclusion:
Both statements are correct.