Find the set of values of 'a' such that `f(x)=ax-sinx` is increasing on R.

Answer

To find the set of values of 'a' such that the function \( f(x) = ax - \sin x \) is increasing on \( \mathbb{R} \), we need to analyze the derivative of the function.

### Step-by-step Solution:

1. **Find the derivative of the function**:
\[ f'(x) = \frac{d}{dx}(ax - \sin x) = a - \cos x \]

2. **Determine when the function is increasing**:
The function \( f(x) \) is increasing when its derivative is greater than zero:
\[ f'(x) > 0 \implies a - \cos x > 0 \]

3. **Rearranging the inequality**:
Rearranging the inequality gives:
\[ a > \cos x \]

4. **Analyze the range of \( \cos x \)**:
The cosine function oscillates between -1 and 1 for all \( x \in \mathbb{R} \):
\[ -1 \leq \cos x \leq 1 \]

5. **Find the condition for 'a'**:
For \( a \) to be greater than \( \cos x \) for all \( x \), \( a \) must be greater than the maximum value of \( \cos x \):
\[ a > 1 \]

6. **Conclusion**:
Therefore, the set of values of \( a \) such that \( f(x) \) is increasing on \( \mathbb{R} \) is:
\[ a \in (1, \infty) \]

### Final Answer:
The set of values of \( a \) such that \( f(x) = ax - \sin x \) is increasing on \( \mathbb{R} \) is \( a > 1 \) or \( a \in (1, \infty) \).

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