Find the range of the following functions: (where {.} and [.] represent fractional part and greatest integer part functions respectively)
`f(x)=sin^(2)xcos^(4)x`

Explanation

To find the range of the function \( f(x) = \sin^2 x \cos^4 x \), we can follow these steps:

### Step 1: Rewrite the function
We start with the function:
\[ f(x) = \sin^2 x \cos^4 x \]
We can express \(\sin^2 x\) in terms of \(\cos^2 x\):
\[ \sin^2 x = 1 - \cos^2 x \]
Thus, we can rewrite \(f(x)\) as:
\[ f(x) = (1 - \cos^2 x) \cos^4 x \]

### Step 2: Expand the function
Now, we expand the function:
\[ f(x) = \cos^4 x - \cos^6 x \]

### Step 3: Define a new variable
Let \(y = \cos^2 x\). The range of \(\cos^2 x\) is from 0 to 1, so:
\[ y \in [0, 1] \]
Now we can rewrite \(f(x)\) in terms of \(y\):
\[ f(x) = y^2 - y^3 \]

### Step 4: Analyze the function \(g(y) = y^2 - y^3\)
We need to find the range of the function \(g(y) = y^2 - y^3\) for \(y \in [0, 1]\).

### Step 5: Find critical points
To find the maximum and minimum values, we take the derivative of \(g(y)\):
\[ g'(y) = 2y - 3y^2 \]
Setting the derivative to zero to find critical points:
\[ 2y - 3y^2 = 0 \]
Factoring gives:
\[ y(2 - 3y) = 0 \]
Thus, \(y = 0\) or \(y = \frac{2}{3}\).

### Step 6: Evaluate \(g(y)\) at critical points and endpoints
Now we evaluate \(g(y)\) at the critical points and endpoints:
1. \(g(0) = 0^2 - 0^3 = 0\)
2. \(g(1) = 1^2 - 1^3 = 0\)
3. \(g\left(\frac{2}{3}\right) = \left(\frac{2}{3}\right)^2 - \left(\frac{2}{3}\right)^3 = \frac{4}{9} - \frac{8}{27}\)

To simplify \(g\left(\frac{2}{3}\right)\):
\[ g\left(\frac{2}{3}\right) = \frac{4}{9} - \frac{8}{27} = \frac{12}{27} - \frac{8}{27} = \frac{4}{27} \]

### Step 7: Determine the range
From the evaluations:
- At \(y = 0\), \(g(0) = 0\)
- At \(y = 1\), \(g(1) = 0\)
- At \(y = \frac{2}{3}\), \(g\left(\frac{2}{3}\right) = \frac{4}{27}\)

Thus, the maximum value of \(f(x)\) is \(\frac{4}{27}\) and the minimum value is \(0\).

### Conclusion
The range of the function \(f(x) = \sin^2 x \cos^4 x\) is:
\[ \text{Range of } f(x) = [0, \frac{4}{27}] \]