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, 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 need to find its derivative \( f'(x) \):

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

Calculating each term, we get:

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

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

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

\[ f'(x) > 0 \]

This leads to the inequality:

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

### Step 3: Analyze the trigonometric terms

The maximum and minimum values of \( \sin(2x) \), \( \sin(x) \), and \( \cos(x) \) must be considered. The ranges are:

- \( \sin(2x) \) varies between -1 and 1.
- \( \sin(x) \) varies between -1 and 1.
- \( \cos(x) \) varies between -1 and 1.

Thus, we can find the maximum and minimum of the expression \( -2\sin(2x) + \cos(x) - \sin(x) \).

### Step 4: Find the maximum value of \( -2\sin(2x) + \cos(x) - \sin(x) \)

The maximum value occurs when \( \sin(2x) = -1 \) (which gives \( 2 \)), \( \cos(x) = 1 \) (which gives \( 1 \)), and \( \sin(x) = -1 \) (which gives \( 1 \)). Therefore, the maximum value is:

\[ -2(-1) + 1 - (-1) = 2 + 1 + 1 = 4 \]

Thus, the minimum value of the expression \( -2\sin(2x) + \cos(x) - \sin(x) \) is \( -4 \).

### Step 5: Set up the inequality for \( a \)

To ensure that \( f'(x) > 0 \):

\[ a - 4 > 0 \implies a > 4 \]

### Step 6: Determine the range of \( a \)

The range of \( a \) is \( (4, \infty) \). We can express this in the form \( \left(\frac{m}{n}, \infty\right) \). Here, \( m = 4 \) and \( n = 1 \).

### Step 7: Calculate \( m - n \)

Now, we find \( m - n \):

\[ m - n = 4 - 1 = 3 \]

Thus, the minimum value of \( m - n \) is:

\[ \boxed{3} \]