If `f(x)=2(7cos x +24 sin x) (7 sinx -24 cos x)`, for even `x in R` then maximum value of f(x) is __________

Explanation

To find the maximum value of the function \( f(x) = 2(7\cos x + 24\sin x)(7\sin x - 24\cos x) \), we can follow these steps:

### Step 1: Rewrite the function
Start by rewriting the function in a more manageable form. We can express \( f(x) \) as:
\[ f(x) = 2(7\cos x + 24\sin x)(7\sin x - 24\cos x) \]

### Step 2: Factor out constants
We can factor out a common term from both parts of the product:
\[ f(x) = 2 \cdot (7\cos x + 24\sin x)(7\sin x - 24\cos x) \]

### Step 3: Multiply and simplify
Next, we will multiply the two expressions:
\[ f(x) = 2 \cdot (7\cos x \cdot 7\sin x - 24\cos^2 x + 24\sin^2 x + 24\sin x \cdot 7\sin x) \]
This simplifies to:
\[ f(x) = 2 \cdot (49\sin x \cos x - 24(\cos^2 x + \sin^2 x)) \]
Using the identity \( \cos^2 x + \sin^2 x = 1 \):
\[ f(x) = 2 \cdot (49\sin x \cos x - 24) \]

### Step 4: Use double angle identity
Using the double angle identity \( \sin(2x) = 2\sin x \cos x \):
\[ f(x) = 49\sin(2x) - 48 \]

### Step 5: Find the maximum value
The maximum value of \( \sin(2x) \) is 1. Thus, we can find the maximum value of \( f(x) \):
\[ f_{\text{max}} = 49 \cdot 1 - 48 = 49 - 48 = 1 \]

### Step 6: Conclusion
Therefore, the maximum value of \( f(x) \) is:
\[ \boxed{1} \]

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