The domain of definition of `y=[log_(10)((5x-x^(2))/(4))]^(1//2)` is

Options

A. [1, 4]
B. [-4, -1]
C. [0, 5]
D. [-1, 5]

Answer

To find the domain of the function \( y = \left[\log_{10}\left(\frac{5x - x^2}{4}\right)\right]^{1/2} \), we need to ensure that the expression inside the logarithm is positive, as the logarithm is only defined for positive values.

### Step-by-step Solution:

1. **Identify the expression inside the logarithm**:
We have \( \frac{5x - x^2}{4} \). For the logarithm to be defined, we need:
\[ \frac{5x - x^2}{4} > 0 \]

2. **Simplify the inequality**:
Since \( 4 \) is a positive constant, we can multiply both sides of the inequality by \( 4 \) without changing the direction of the inequality:
\[ 5x - x^2 > 0 \]

3. **Rearrange the inequality**:
Rearranging gives us:
\[ -x^2 + 5x > 0 \]
or equivalently,
\[ x^2 - 5x < 0 \]

4. **Factor the quadratic expression**:
We can factor the left-hand side:
\[ x(x - 5) < 0 \]

5. **Find the critical points**:
The critical points are found by setting the factors equal to zero:
\[ x = 0 \quad \text{and} \quad x = 5 \]

6. **Test intervals**:
We need to test the intervals determined by the critical points \( 0 \) and \( 5 \):
- For \( x < 0 \) (e.g., \( x = -1 \)): \( (-1)(-6) > 0 \) (not valid)
- For \( 0 < x < 5 \) (e.g., \( x = 1 \)): \( (1)(-4) < 0 \) (valid)
- For \( x > 5 \) (e.g., \( x = 6 \)): \( (6)(1) > 0 \) (not valid)

7. **Conclusion**:
The valid interval for \( x \) is \( (0, 5) \). Therefore, the domain of the function is:
\[ (0, 5) \]

### Final Answer:
The domain of the function \( y = \left[\log_{10}\left(\frac{5x - x^2}{4}\right)\right]^{1/2} \) is \( (0, 5) \).