Find the mimimum value of sin x cos x.

Answer

To find the minimum value of \( f(x) = \sin x \cos x \), we can follow these steps:

### Step 1: Rewrite the function
We can use the double angle identity for sine, which states that:
\[ \sin(2x) = 2 \sin x \cos x \]
Thus, we can express \( f(x) \) as:
\[ f(x) = \sin x \cos x = \frac{1}{2} \sin(2x) \]

### Step 2: Determine the range of \( \sin(2x) \)
The sine function has a range of \([-1, 1]\). Therefore, the range of \( \sin(2x) \) is also \([-1, 1]\).

### Step 3: Find the range of \( f(x) \)
Since \( f(x) = \frac{1}{2} \sin(2x) \), we can find the range of \( f(x) \) by multiplying the range of \( \sin(2x) \) by \(\frac{1}{2}\):
\[ \text{Range of } f(x) = \left[-\frac{1}{2}, \frac{1}{2}\right] \]

### Step 4: Identify the minimum value
From the range we found in Step 3, the minimum value of \( f(x) \) is:
\[ -\frac{1}{2} \]

### Conclusion
Thus, the minimum value of \( \sin x \cos x \) is:
\[ \boxed{-\frac{1}{2}} \]