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:
\[ y = 1 + 8 \sin^2(x^2) \cos^2(x^2) \]

### Step 2: Use the double angle identity
Recall that \( \sin^2(x) \cos^2(x) = \frac{1}{4} \sin^2(2x) \). Therefore, we can rewrite \(8 \sin^2(x^2) \cos^2(x^2)\) as:
\[ 8 \sin^2(x^2) \cos^2(x^2) = 8 \cdot \frac{1}{4} \sin^2(2x^2) = 2 \sin^2(2x^2) \]
Now, substitute this back into the expression:
\[ y = 1 + 2 \sin^2(2x^2) \]

### Step 3: Determine the maximum value of \(\sin^2(2x^2)\)
The maximum value of \(\sin^2(2x^2)\) is 1, since the sine function oscillates between -1 and 1. Thus:
\[ \sin^2(2x^2) \leq 1 \]

### Step 4: Substitute the maximum value back into the expression
Substituting this maximum value into our expression for \(y\):
\[ y = 1 + 2 \cdot 1 = 1 + 2 = 3 \]

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

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