The domain of definition of `f(x)=log_(1.7)((2-phi'(x))/(x+1))^(1//2)`, where ` phi(x)=(x^(3))/(3)-(3)/(2)x^(2)-2x+(3)/(2)`, is

Options

A. `(-oo,-4)`
B. `(-4,oo)`
C. `(-oo,-1) cup (-1,4)`
D. `(-oo,-1) cup (-1,4)]`

Answer

To find the domain of the function \( f(x) = \log_{1.7}\left(\frac{2 - \phi'(x)}{(x + 1)^{1/2}}\right) \), where \( \phi(x) = \frac{x^3}{3} - \frac{3}{2}x^2 - 2x + \frac{3}{2} \), we need to ensure that the argument of the logarithm is positive.

### Step-by-Step Solution:

1. **Differentiate \( \phi(x) \)**:
\[ \phi'(x) = \frac{d}{dx}\left(\frac{x^3}{3} - \frac{3}{2}x^2 - 2x + \frac{3}{2}\right) \]
Using the power rule:
\[ \phi'(x) = x^2 - 3x - 2 \]

2. **Set up the inequality for the logarithm**:
The argument of the logarithm must be greater than zero:
\[ \frac{2 - \phi'(x)}{(x + 1)^{1/2}} > 0 \]
This implies:
\[ 2 - \phi'(x) > 0 \quad \text{and} \quad (x + 1)^{1/2} > 0 \]

3. **Solve the first inequality**:
\[ 2 - (x^2 - 3x - 2) > 0 \]
Simplifying:
\[ 2 - x^2 + 3x + 2 > 0 \]
\[ -x^2 + 3x + 4 > 0 \]
Multiplying through by -1 (and reversing the inequality):
\[ x^2 - 3x - 4 < 0 \]

4. **Factor the quadratic**:
\[ (x - 4)(x + 1) < 0 \]

5. **Determine the intervals**:
The roots of the equation \( (x - 4)(x + 1) = 0 \) are \( x = 4 \) and \( x = -1 \). We analyze the sign of the product in the intervals:
- For \( x < -1 \): both factors are negative, product is positive.
- For \( -1 < x < 4 \): one factor is negative, the other is positive, product is negative.
- For \( x > 4 \): both factors are positive, product is positive.

Thus, the solution to \( (x - 4)(x + 1) < 0 \) is:
\[ -1 < x < 4 \]

6. **Consider the second inequality**:
The term \( (x + 1)^{1/2} > 0 \) implies:
\[ x + 1 > 0 \quad \Rightarrow \quad x > -1 \]

7. **Combine the conditions**:
From the inequalities, we have:
\[ -1 < x < 4 \]
However, we must exclude \( x = -1 \) since it makes the denominator zero.

8. **Final domain**:
The domain of \( f(x) \) is:
\[ (-1, 4) \]

### Conclusion:
The domain of the function \( f(x) \) is \( (-1, 4) \).