Prove that the following functions are increasing for the given intervals,
(i) `y =e^(x) + sinx, x epsilon R^(+)`
(ii) `y=sin x+ tan x- 2x, x epsilon(0, pi/2)`
(iii) ` y = x + sin x , x epsilon R`.

Answer

To prove that the given functions are increasing in their respective intervals, we will find the derivative of each function and show that the derivative is positive in the specified intervals.

### (i) For the function \( y = e^x + \sin x \), \( x \in \mathbb{R}^+ \):

1. **Find the derivative**:
\[ y' = \frac{d}{dx}(e^x + \sin x) = e^x + \cos x \]

2. **Analyze the derivative**:
- Since \( e^x > 0 \) for all \( x \) and \( \cos x \) oscillates between -1 and 1, we have:
\[ y' = e^x + \cos x > 0 \quad \text{for all } x \in \mathbb{R}^+ \]
- Therefore, \( y' > 0 \) implies that \( y \) is an increasing function for \( x \in \mathbb{R}^+ \).

### (ii) For the function \( y = \sin x + \tan x - 2x \), \( x \in (0, \frac{\pi}{2}) \):

1. **Find the derivative**:
\[ y' = \frac{d}{dx}(\sin x + \tan x - 2x) = \cos x + \sec^2 x - 2 \]

2. **Rewrite the derivative**:
\[ y' = \cos x + \frac{1}{\cos^2 x} - 2 \]
- Let \( g(x) = \cos x + \sec^2 x - 2 \).

3. **Analyze the derivative**:
- We can find a common denominator:
\[ g(x) = \frac{\cos^3 x + 1 - 2\cos^2 x}{\cos^2 x} \]
- The numerator simplifies to:
\[ g(x) = \frac{(\cos x - 1)(\cos^2 x + \cos x + 1)}{\cos^2 x} \]
- Since \( \cos x \) is positive in \( (0, \frac{\pi}{2}) \) and \( \cos^2 x + \cos x + 1 > 0 \), we conclude that \( g(x) > 0 \) in this interval.

4. **Conclusion**:
- Therefore, \( y' > 0 \) implies that \( y \) is an increasing function for \( x \in (0, \frac{\pi}{2}) \).

### (iii) For the function \( y = x + \sin x \), \( x \in \mathbb{R} \):

1. **Find the derivative**:
\[ y' = \frac{d}{dx}(x + \sin x) = 1 + \cos x \]

2. **Analyze the derivative**:
- The cosine function oscillates between -1 and 1:
\[ y' = 1 + \cos x \geq 0 \quad \text{for all } x \in \mathbb{R} \]
- Since \( 1 + \cos x \) is always greater than or equal to 0, we have \( y' > 0 \) for all \( x \).

3. **Conclusion**:
- Therefore, \( y \) is an increasing function for \( x \in \mathbb{R} \).

### Summary of Results:
- The function \( y = e^x + \sin x \) is increasing for \( x \in \mathbb{R}^+ \).
- The function \( y = \sin x + \tan x - 2x \) is increasing for \( x \in (0, \frac{\pi}{2}) \).
- The function \( y = x + \sin x \) is increasing for \( x \in \mathbb{R} \).