The following functions are defined for any two distinct, non zero integers a and b
`f_(1)(a, b)=|a|xxb^(2)`
`f_(2)(a, b)=a^(2)xx|b|`
`f_(3)(a, b)=((a^(2)+b^(2)))/(2)`
`f_(4)(a, b)=((a)/(2)+(b)/(2))`
Which one of the following is necessarily greater than zero?

Options

A. A. `f_(1)(a, b)xxf_(2)(a, b)xxf_(3)(a, b)`
B. B. `f_(1)(a, b)xxf_(2)(a, b)xxf_(4)(a, b)`
C. C. `f_(1)(a, b)xxf_(3)(a, b)xxf_(4)(a, b)`
D. D. `f_(2)(a, b)xxf_(3)(a, b)xxf_(4)(a, b)`

Answer

To determine which of the functions \( f_1(a, b) \), \( f_2(a, b) \), \( f_3(a, b) \), and \( f_4(a, b) \) is necessarily greater than zero for any two distinct, non-zero integers \( a \) and \( b \), we will analyze each function step by step.

### Step 1: Analyze \( f_1(a, b) = |a| \cdot b^2 \)

- **Calculation**:
- The absolute value \( |a| \) is always positive since \( a \) is a non-zero integer.
- The square \( b^2 \) is also always positive since squaring any non-zero integer results in a positive value.
- Therefore, \( f_1(a, b) = |a| \cdot b^2 \) is the product of two positive numbers, which is always positive.

### Step 2: Analyze \( f_2(a, b) = a^2 \cdot |b| \)

- **Calculation**:
- The square \( a^2 \) is always positive since \( a \) is a non-zero integer.
- The absolute value \( |b| \) is also always positive since \( b \) is a non-zero integer.
- Thus, \( f_2(a, b) = a^2 \cdot |b| \) is the product of two positive numbers, which is always positive.

### Step 3: Analyze \( f_3(a, b) = \frac{a^2 + b^2}{2} \)

- **Calculation**:
- Both \( a^2 \) and \( b^2 \) are always positive since \( a \) and \( b \) are non-zero integers.
- Therefore, \( a^2 + b^2 \) is positive.
- Dividing a positive number by 2 still results in a positive number.
- Hence, \( f_3(a, b) = \frac{a^2 + b^2}{2} \) is always positive.

### Step 4: Analyze \( f_4(a, b) = \frac{a}{2} + \frac{b}{2} \)

- **Calculation**:
- The values of \( a \) and \( b \) can be either positive or negative integers.
- For example, if \( a = -4 \) and \( b = 2 \):
\[ f_4(-4, 2) = \frac{-4}{2} + \frac{2}{2} = -2 + 1 = -1 \]
- This shows that \( f_4(a, b) \) can be negative depending on the values of \( a \) and \( b \).
- Therefore, \( f_4(a, b) \) is not necessarily greater than zero.

### Conclusion

From the analysis, we find that:

- \( f_1(a, b) \) is always positive.
- \( f_2(a, b) \) is always positive.
- \( f_3(a, b) \) is always positive.
- \( f_4(a, b) \) can be negative.

Thus, the functions \( f_1(a, b) \), \( f_2(a, b) \), and \( f_3(a, b) \) are necessarily greater than zero, but the question asks for which one is necessarily greater than zero. Since we are looking for a single function, we can conclude that **all three functions \( f_1, f_2, \) and \( f_3 \) are correct, but if we have to choose one, we can select \( f_1 \)**.