Find `max^(m)&min^(m)` value of `sin^(4)theta.cos^(4)theta`

Answer

To find the maximum and minimum values of the expression \( \sin^4 \theta \cos^4 \theta \), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ y = \sin^4 \theta \cos^4 \theta \]
To simplify, we can rewrite it using the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \):
\[ y = \left( \sin^2 \theta \cos^2 \theta \right)^2 \]
Now, using the identity \( \sin^2 \theta \cos^2 \theta = \frac{1}{4} \sin^2 2\theta \):
\[ y = \left( \frac{1}{4} \sin^2 2\theta \right)^2 = \frac{1}{16} \sin^4 2\theta \]

### Step 2: Find Maximum Value
The maximum value of \( \sin^2 x \) is 1, thus:
\[ \text{Maximum of } \sin^4 2\theta = 1^2 = 1 \]
Therefore, the maximum value of \( y \) is:
\[ \text{Maximum of } y = \frac{1}{16} \cdot 1 = \frac{1}{16} \]

### Step 3: Find Minimum Value
The minimum value of \( \sin^2 x \) is 0, thus:
\[ \text{Minimum of } \sin^4 2\theta = 0^2 = 0 \]
Therefore, the minimum value of \( y \) is:
\[ \text{Minimum of } y = \frac{1}{16} \cdot 0 = 0 \]

### Conclusion
The maximum and minimum values of \( \sin^4 \theta \cos^4 \theta \) are:
\[ \text{Max}(y) = \frac{1}{16}, \quad \text{Min}(y) = 0 \]