For a polynomia g(x) with real coefficients, let `m_(g)` denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficients defined by
`S={(x^(2)-1)^(2)(a_(0)+a_(1)+a_(2)x^(2)+a_(3)x^(3)): a_(0) a_(1), a_(2), a_(3)= RR}`
For a polynomial f. let f and f denote its first and second order derivties, respectively. Then the minimum possible value of `(m_(f')+m_(f"))`. where ` f in S` is, ____

Answer

To solve the problem, we need to analyze the polynomial \( f(x) \) defined in the set \( S \) and calculate the minimum possible value of \( m(f') + m(f'') \), where \( m(g) \) denotes the number of distinct real roots of the polynomial \( g(x) \).

### Step 1: Define the polynomial \( f(x) \)
The polynomial \( f(x) \) is given by:
\[ f(x) = (x^2 - 1)^2 (a_0 + a_1 x + a_2 x^2 + a_3 x^3) \]
where \( a_0, a_1, a_2, a_3 \) are real coefficients.

### Step 2: Identify the roots of \( f(x) \)
The term \( (x^2 - 1)^2 \) contributes roots at \( x = 1 \) and \( x = -1 \), each with multiplicity 2. Therefore, \( f(x) \) has at least 2 distinct real roots from this factor.

The polynomial \( a_0 + a_1 x + a_2 x^2 + a_3 x^3 \) is a cubic polynomial, which can have at most 3 distinct real roots. Thus, the total number of distinct real roots of \( f(x) \) can be at most 5 (2 from \( (x^2 - 1)^2 \) and up to 3 from the cubic polynomial).

### Step 3: Calculate the first derivative \( f'(x) \)
Using the product rule and chain rule, we differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}[(x^2 - 1)^2] \cdot (a_0 + a_1 x + a_2 x^2 + a_3 x^3) + (x^2 - 1)^2 \cdot \frac{d}{dx}[a_0 + a_1 x + a_2 x^2 + a_3 x^3] \]

Calculating the derivative of \( (x^2 - 1)^2 \):
\[ \frac{d}{dx}[(x^2 - 1)^2] = 2(x^2 - 1)(2x) = 4x(x^2 - 1) \]

The derivative of the cubic polynomial is:
\[ \frac{d}{dx}[a_0 + a_1 x + a_2 x^2 + a_3 x^3] = a_1 + 2a_2 x + 3a_3 x^2 \]

Thus, we have:
\[ f'(x) = 4x(x^2 - 1)(a_0 + a_1 x + a_2 x^2 + a_3 x^3) + (x^2 - 1)^2(a_1 + 2a_2 x + 3a_3 x^2) \]

### Step 4: Analyze \( m(f') \)
The polynomial \( f'(x) \) is a product of \( (x^2 - 1) \) and other terms. Since \( (x^2 - 1) \) contributes roots at \( x = 1 \) and \( x = -1 \), \( f'(x) \) will have at least these two roots. The behavior of the other polynomial factors will determine if there are additional distinct roots.

### Step 5: Calculate the second derivative \( f''(x) \)
Differentiating \( f'(x) \) will yield \( f''(x) \). The degree of \( f'(x) \) is less than that of \( f(x) \), which means \( f''(x) \) will have a degree that is 2 less than \( f(x) \).

### Step 6: Analyze \( m(f'') \)
Since \( f(x) \) is of degree 6, \( f'(x) \) will be of degree 5, and \( f''(x) \) will be of degree 4. A polynomial of degree 4 can have at most 4 distinct real roots. However, since \( f'(x) \) has already contributed 2 distinct roots, \( f''(x) \) will typically have at least 1 distinct root.

### Step 7: Calculate \( m(f') + m(f'') \)
From our analysis:
- \( m(f') \) can be at least 2.
- \( m(f'') \) can be at least 1.

Thus, the minimum possible value of \( m(f') + m(f'') \) is:
\[ m(f') + m(f'') = 2 + 1 = 3 \]

### Final Answer
The minimum possible value of \( m(f') + m(f'') \) is **3**.