The maximum value of `1+8sin^(2)x^(2)cos^(2)x^(2)` is

Options

A. 3
B. `-1`
C. `-8`
D. 9

Answer

To find the maximum value of the expression \(1 + 8 \sin^2(x^2) \cos^2(x^2)\), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ f(x) = 1 + 8 \sin^2(x^2) \cos^2(x^2) \]
We can rewrite \(8 \sin^2(x^2) \cos^2(x^2)\) using the identity \( \sin^2(2\theta) = 4 \sin^2(\theta) \cos^2(\theta) \):
\[ f(x) = 1 + 2 \cdot 4 \sin^2(x^2) \cos^2(x^2) = 1 + 2 \sin^2(2x^2) \]

### Step 2: Determine the Range of \( \sin^2(2x^2) \)
The function \( \sin^2(2x^2) \) oscillates between 0 and 1. Therefore, we can express the range of \(2 \sin^2(2x^2)\):
\[ 0 \leq \sin^2(2x^2) \leq 1 \]
This implies:
\[ 0 \leq 2 \sin^2(2x^2) \leq 2 \]

### Step 3: Find the Maximum Value of \( f(x) \)
Now, substituting the maximum value of \(2 \sin^2(2x^2)\) into the expression for \(f(x)\):
\[ f(x) = 1 + 2 \sin^2(2x^2) \]
The maximum value occurs when \(\sin^2(2x^2) = 1\):
\[ f(x) = 1 + 2 \cdot 1 = 1 + 2 = 3 \]

### Conclusion
Thus, the maximum value of the expression \(1 + 8 \sin^2(x^2) \cos^2(x^2)\) is:
\[ \boxed{3} \]