The domain of definition of the function
` f(x) = sqrt(log_(10) ((5 x -x^2)/(4))) ` is

Options

A. `[1,4]`
B. `(1,4)`
C. `(0,5)`
D. `[0,5]`

Answer

To find the domain of the function \( f(x) = \sqrt{\log_{10} \left( \frac{5x - x^2}{4} \right)} \), we need to ensure that the expression inside the square root is non-negative. This leads us to the following steps:

### Step 1: Set the inequality for the logarithm
Since the square root function requires its argument to be non-negative, we need:
\[ \log_{10} \left( \frac{5x - x^2}{4} \right) \geq 0 \]

### Step 2: Convert the logarithmic inequality
The logarithm is non-negative when its argument is greater than or equal to 1:
\[ \frac{5x - x^2}{4} \geq 1 \]

### Step 3: Clear the fraction
Multiply both sides of the inequality by 4 (since 4 is positive, the direction of the inequality remains unchanged):
\[ 5x - x^2 \geq 4 \]

### Step 4: Rearrange the inequality
Rearranging gives:
\[ -x^2 + 5x - 4 \geq 0 \]
or equivalently,
\[ x^2 - 5x + 4 \leq 0 \]

### Step 5: Factor the quadratic
Now, we factor the quadratic expression:
\[ x^2 - 5x + 4 = (x - 1)(x - 4) \]
Thus, we need to solve:
\[ (x - 1)(x - 4) \leq 0 \]

### Step 6: Determine the intervals
To find the intervals where this inequality holds, we identify the roots of the equation \( (x - 1)(x - 4) = 0 \), which are \( x = 1 \) and \( x = 4 \). We analyze the sign of the product in the intervals:
- \( (-\infty, 1) \)
- \( (1, 4) \)
- \( (4, \infty) \)

### Step 7: Test the intervals
1. For \( x < 1 \) (e.g., \( x = 0 \)):
\[ (0 - 1)(0 - 4) = ( -1)( -4) = 4 \quad (\text{positive}) \]
2. For \( 1 < x < 4 \) (e.g., \( x = 2 \)):
\[ (2 - 1)(2 - 4) = (1)( -2) = -2 \quad (\text{negative}) \]
3. For \( x > 4 \) (e.g., \( x = 5 \)):
\[ (5 - 1)(5 - 4) = (4)(1) = 4 \quad (\text{positive}) \]

### Step 8: Conclusion
The inequality \( (x - 1)(x - 4) \leq 0 \) is satisfied in the interval:
\[ [1, 4] \]
Thus, the domain of the function \( f(x) \) is:
\[ \boxed{[1, 4]} \]