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

Options

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

Explanation

To determine whether the statements regarding the functions are correct, we need to analyze each function by finding its derivative and checking the intervals where the derivative is positive.

### Step 1: Analyze the first function
The first function is given by:
\[ y_1 = \frac{e^x + e^{-x}}{2} \]

**Step 1.1: Find the derivative**
To check if this function is increasing on \([0, \infty)\), we need to find the derivative \( \frac{dy_1}{dx} \):
\[ \frac{dy_1}{dx} = \frac{1}{2} \left( e^x - e^{-x} \right) \]

**Step 1.2: Set the derivative greater than zero**
To determine where this function is increasing, we set the derivative greater than zero:
\[ \frac{1}{2} (e^x - e^{-x}) > 0 \]
This simplifies to:
\[ e^x - e^{-x} > 0 \]
Multiplying through by \( e^x \) (which is always positive):
\[ e^{2x} - 1 > 0 \]
This implies:
\[ e^{2x} > 1 \implies 2x > 0 \implies x > 0 \]

**Step 1.3: Conclusion for the first function**
Thus, \( \frac{dy_1}{dx} > 0 \) for \( x > 0 \), and therefore \( y_1 \) is an increasing function on \([0, \infty)\).

### Step 2: Analyze the second function
The second function is given by:
\[ y_2 = \frac{e^x - e^{-x}}{2} \]

**Step 2.1: Find the derivative**
To check if this function is increasing on \((- \infty, \infty)\), we find the derivative \( \frac{dy_2}{dx} \):
\[ \frac{dy_2}{dx} = \frac{1}{2} \left( e^x + e^{-x} \right) \]

**Step 2.2: Set the derivative greater than zero**
To determine where this function is increasing, we set the derivative greater than zero:
\[ \frac{1}{2} (e^x + e^{-x}) > 0 \]
Since \( e^x \) and \( e^{-x} \) are both positive for all \( x \), we have:
\[ e^x + e^{-x} > 0 \]
This is always true for all \( x \).

**Step 2.3: Conclusion for the second function**
Thus, \( \frac{dy_2}{dx} > 0 \) for all \( x \), and therefore \( y_2 \) is an increasing function on \((- \infty, \infty)\).

### Final Conclusion
Both statements are correct:
1. \( y_1 = \frac{e^x + e^{-x}}{2} \) is an increasing function on \([0, \infty)\).
2. \( y_2 = \frac{e^x - e^{-x}}{2} \) is an increasing function on \((- \infty, \infty)\).

### Answer
Both statements are correct.

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