Let `f (x) = ax+cos 2x +sin x+ cos x ` is defined for `AA x in R and a in R` and is strictely increasing function. If the range of a is `[(m)/(n),oo),` then find the minimum vlaue of `(m- n).`

Answer

To solve the problem step by step, we need to analyze the function \( f(x) = ax + \cos(2x) + \sin(x) + \cos(x) \) and determine the conditions under which it is strictly increasing.

### Step 1: Find the derivative of \( f(x) \)

To determine if \( f(x) \) is strictly increasing, we first find its derivative:

\[ f'(x) = \frac{d}{dx}(ax + \cos(2x) + \sin(x) + \cos(x)) \]

Calculating the derivative, we have:

\[ f'(x) = a - 2\sin(2x) + \cos(x) - \sin(x) \]

### Step 2: Set the condition for strict increase

For \( f(x) \) to be strictly increasing, we need:

\[ f'(x) > 0 \]

This gives us the inequality:

\[ a - 2\sin(2x) + \cos(x) - \sin(x) > 0 \]

Rearranging this, we find:

\[ a > 2\sin(2x) + \sin(x) - \cos(x) \]

### Step 3: Define \( g(x) \)

Let’s define:

\[ g(x) = 2\sin(2x) + \sin(x) - \cos(x) \]

To find the minimum value of \( a \), we need to find the maximum value of \( g(x) \).

### Step 4: Analyze \( g(x) \)

We can rewrite \( g(x) \):

\[ g(x) = 2(2\sin(x)\cos(x)) + \sin(x) - \cos(x) \]

### Step 5: Find critical points of \( g(x) \)

To find the maximum of \( g(x) \), we take the derivative \( g'(x) \) and set it to zero:

\[ g'(x) = 4\cos(2x) + \cos(x) + \sin(x) \]

Setting \( g'(x) = 0 \) will give us the critical points, but we can also analyze the behavior of \( g(x) \) directly.

### Step 6: Determine the maximum value of \( g(x) \)

To find the maximum value of \( g(x) \), we can evaluate it at specific points or use known values of trigonometric functions.

1. **At \( x = 0 \)**:
\[ g(0) = 2\sin(0) + \sin(0) - \cos(0) = 0 - 1 = -1 \]

2. **At \( x = \frac{\pi}{2} \)**:
\[ g\left(\frac{\pi}{2}\right) = 2\sin(\pi) + \sin\left(\frac{\pi}{2}\right) - \cos\left(\frac{\pi}{2}\right) = 0 + 1 - 0 = 1 \]

3. **At \( x = \frac{\pi}{4} \)**:
\[ g\left(\frac{\pi}{4}\right) = 2\sin\left(\frac{\pi}{2}\right) + \sin\left(\frac{\pi}{4}\right) - \cos\left(\frac{\pi}{4}\right) = 2 + \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = 2 \]

### Step 7: Maximum value of \( g(x) \)

From our evaluations, we find that the maximum value of \( g(x) \) is \( 2 \).

### Step 8: Find the minimum value of \( a \)

Thus, we have:

\[ a > g(x)_{\text{max}} = 2 \]

### Step 9: Determine the range of \( a \)

The range of \( a \) is \( \left(\frac{17}{8}, \infty\right) \). Here, \( m = 17 \) and \( n = 8 \).

### Step 10: Calculate \( m - n \)

Finally, we compute:

\[ m - n = 17 - 8 = 9 \]

### Final Answer

The minimum value of \( m - n \) is \( \boxed{9} \).