Find the values of 'a' for which `f(x)=sinx-ax+b` is decreasing function on R.

Answer

To find the values of 'a' for which the function \( f(x) = \sin x - ax + b \) is a decreasing function on \( \mathbb{R} \), we will follow these steps:

### Step 1: Differentiate the function
We start by finding the derivative of the function \( f(x) \).

\[ f'(x) = \frac{d}{dx}(\sin x - ax + b) \]

Using the rules of differentiation, we know:
- The derivative of \( \sin x \) is \( \cos x \).
- The derivative of \( -ax \) is \( -a \).
- The derivative of \( b \) (a constant) is \( 0 \).

Thus, we have:

\[ f'(x) = \cos x - a \]

### Step 2: Set the condition for decreasing function
For the function to be decreasing, we need:

\[ f'(x) \leq 0 \]

Substituting our expression for \( f'(x) \):

\[ \cos x - a \leq 0 \]

This can be rearranged to:

\[ \cos x \leq a \]

### Step 3: Analyze the maximum value of \( \cos x \)
The cosine function oscillates between -1 and 1 for all \( x \). Therefore, the maximum value of \( \cos x \) is 1. For the inequality \( \cos x \leq a \) to hold for all \( x \), we must have:

\[ a \geq 1 \]

### Step 4: Conclusion
Thus, the values of \( a \) for which the function \( f(x) \) is a decreasing function on \( \mathbb{R} \) are:

\[ a \geq 1 \]

### Summary
The function \( f(x) = \sin x - ax + b \) is decreasing on \( \mathbb{R} \) when \( a \) is greater than or equal to 1.

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