Consider the polynomial f`(x)= 1+2x+3x^2+4x^3`. Let s be the sum of all distinct real roots of `f(x)`and let `t= |s|`.

Options

A. increasing in `(-t, 1/4)` and decreasing in `(-(1)/(4),t)`
B. decreasing in `(-t, -1/4)` and increasing in `(-(1)/(4),t)`
C. increasing in `(-t,t)`
D. decreasing in `(-t,t)`

Answer

To solve the problem, we need to find the polynomial \( f(x) \) from its derivative \( f'(x) = 1 + 2x + 3x^2 + 4x^3 \), and then determine the sum of all distinct real roots of \( f(x) \).

### Step 1: Find the polynomial \( f(x) \)

Given:
\[ f'(x) = 1 + 2x + 3x^2 + 4x^3 \]

To find \( f(x) \), we integrate \( f'(x) \):
\[ f(x) = \int (1 + 2x + 3x^2 + 4x^3) \, dx \]

Calculating the integral:
\[ f(x) = x + x^2 + x^3 + x^4 + C \]
where \( C \) is the constant of integration.

### Step 2: Determine the roots of \( f(x) \)

Now, we need to find the roots of:
\[ f(x) = x + x^2 + x^3 + x^4 + C = 0 \]

### Step 3: Analyze the behavior of \( f'(x) \)

Next, we analyze \( f'(x) \):
\[ f'(x) = 1 + 2x + 3x^2 + 4x^3 \]

To find the critical points, we set \( f'(x) = 0 \):
\[ 4x^3 + 3x^2 + 2x + 1 = 0 \]

### Step 4: Check the discriminant of \( f'(x) \)

The discriminant \( D \) of the cubic polynomial can help us determine the nature of the roots:
\[ D = b^2 - 4ac \]
For \( 4x^3 + 3x^2 + 2x + 1 \), we can use numerical methods or graphing to check for real roots. However, we can also analyze the behavior of \( f'(x) \).

### Step 5: Determine the intervals of increase and decrease

Since \( f'(x) \) is a cubic polynomial, we can analyze its sign:
- If \( f'(x) > 0 \), then \( f(x) \) is increasing.
- If \( f'(x) < 0 \), then \( f(x) \) is decreasing.

Given that the discriminant is negative, \( f'(x) \) has no real roots and is always positive (since the leading coefficient is positive). Thus, \( f(x) \) is strictly increasing.

### Step 6: Roots of \( f(x) \)

Since \( f(x) \) is strictly increasing, it can have at most one real root. The sum of all distinct real roots \( s \) is simply the single root (if it exists).

### Step 7: Calculate \( t = |s| \)

Since \( f(x) \) is strictly increasing, it will cross the x-axis at most once. Thus, we can conclude:
- If \( C \) is chosen such that \( f(0) = C \) is positive, there are no real roots.
- If \( C \) is negative, there is exactly one real root.

Thus, \( t = |s| \) will be the absolute value of that root.

### Conclusion

The final answer depends on the value of \( C \). If \( C \) is negative, \( t \) will be the absolute value of the root. If \( C \) is positive, \( t = 0 \).